Limits, continuity, and convergence

The threshold H(x)=0 for x<0 and one for x≥0 has left limit zero and right limit one at zero, so no two-sided limit. Its chosen value at zero cannot fix that jump. For 1/x at zero, left values tend to negative infinity and right values to positive infinity; there is no common finite or signed-infinite two-sided limit. An infinite limit means every magnitude threshold is eventually crossed in the stated direction, rather than that infinity is an ordinary real value returned at the point.

Limits at infinity describe large inputs instead of small distances to a finite point: f(x)→L as x→+∞ means for every ε>0 there is M such that all domain x>M have |f(x)−L|<ε. For 1/x, choosing M>1/ε proves the zero limit. This resembles the sequence argument but covers all sufficiently large domain inputs, not only positive integers. A sampled integer trend need not determine the behaviour between those inputs.

Algebraic limit laws require that the constituent limits exist in a shared nearby domain. If f→L,g→M, then f+g→L+M, and products tend to LM. The sum follows by assigning each error ε/2. For the product, write fg−LM=f(g−M)+M(f−L). First restrict |f−L|<1 so |f|≤|L|+1, then make both remaining errors small enough to bound the two terms. This local bound is why the argument works even though f varies with x. Differences and fixed scalar multiples follow similarly.

Quotients require M≠0 and a sufficiently small neighbourhood where g stays nonzero. From |g−M|<|M|/2, we obtain |g|>|M|/2. The reciprocal error |1/g−1/M|=|g−M|/(|g||M|) can then be bounded, establishing the quotient law through multiplication. A zero denominator limit is not permission to divide limits; a 0/0 form requires further analysis and can produce different results for different functions.

The squeeze principle follows from inequalities: if lower and upper functions both approach L and eventually enclose f, their epsilon bounds enclose f too. The same argument applies to sequences. It supplies a useful proof when exact values oscillate but their amplitude shrinks, such as |sin x|/n≤1/n for any selected bounded sine factors. The shrinking bound, not the oscillation’s detailed phase, establishes the limit.

Function limits can be disproved by two domain sequences tending to a but giving different value limits. A valid function limit would force every such sequence’s values into the same epsilon bands. The converse criterion also holds: if the epsilon–delta definition failed, one could choose a violating domain point within 1/n for each n and obtain a counterexample sequence. This gives a precise bridge between sequence reasoning and function limits; a few selected approach paths are still insufficient to prove the universal claim.

Epsilon is the requested output accuracy, and delta is a sufficient input neighbourhood. To prove a limit, accept an arbitrary positive ε, choose δ possibly depending on it and fixed problem constants, then show the implication for every eligible x. Delta cannot depend on a favourable x chosen afterwards. One delta for one requested accuracy establishes only that particular implication; a limit requires a rule or argument for every positive accuracy.

For a nonlinear example x²→4 at x→2, factoring gives |x²−4|=|x−2||x+2|. Control the second factor first by requiring |x−2|<1, which puts x between one and three and gives |x+2|<5. Then choose δ=min(1,ε/5). Every eligible x has output error below5δ≤ε. Using the minimum keeps both requirements true. Guessing δ=ε without bounding the extra factor would leave the proof incomplete, even if a few samples happened to fit the band.

Quantifier order explains why a discontinuity cannot be repaired by shrinking a window on just one side. For the threshold at zero, ε=1/3 defeats every proposed limit L and neighbourhood when both sides are available. The outputs zero and one cannot both lie within1/3 of the same L, since that would imply their distance one is less than2/3. Every δ contains a negative and a positive point, so some eligible input violates the requested band. This is a direct negation argument: one output accuracy, then a bad input for every neighbourhood.

The puncture also makes an isolated bad value compatible with a limit. Altering only f(a) leaves all conditions with x≠a unchanged. Changing values on arbitrarily close neighbouring inputs can instead change or destroy the limit. A proof should identify which points the universal implication covers, especially for domains such as positive reals or a discrete subset approaching a boundary.

For a reciprocal near a nonzero a, use a local input restriction to keep |x| away from zero: |x−a|<|a|/2 implies |x|>|a|/2. Then |1/x−1/a|<2|x−a|/|a|². Choosing δ=min(|a|/2,ε|a|²/2) proves the reciprocal limit. This pattern—bound an awkward factor, then allocate an error budget—reappears in continuity, derivative and numerical-error arguments.

The explorer can visualise a band and inspect selected points, but its finite samples are not themselves the proof. The linear and quadratic cases display the algebraic bound governing the entire window. A sufficient bound can be conservative; exceeding it does not automatically disprove a limit or even the current window. A failure needs an actual witness or a separate exact maximum calculation. Distinguish a theorem’s sufficient condition from a necessary condition before interpreting a red or green numerical indicator.

A function is continuous at a domain point a when arbitrarily small output errors can be guaranteed by sufficiently small input errors, including the point itself. In symbols, for every ε>0 there exists δ>0 such that every domain x with |x−a|<δ satisfies |f(x)−f(a)|<ε. At an accumulation point this is equivalent to the punctured limit existing and equalling f(a). At an isolated domain point continuity holds automatically by choosing a neighbourhood containing no other domain points. That does not create a meaningful punctured limit there. At an interval endpoint continuity is relative to the available one-sided domain.

Continuity of a composition needs compatible domains. Suppose f is continuous at a, f(a)=b, and g is continuous at b, with f mapping the relevant neighbourhood into g’s domain. Given an output ε, continuity of g supplies an intermediate tolerance η>0 around b. Continuity of f then supplies δ so |x−a|<δ implies |f(x)−b|<η. Applying g’s condition yields |g(f(x))−g(b)|<ε. Notice that f(x)=b is permitted. An argument using only a punctured limit for g could fail when inner values hit b and g(b) is assigned a different value. Domain and point-value conditions are doing real work here.

The algebraic limit rules establish continuity of sums and products of continuous functions, and of quotients wherever their denominators are nonzero. Polynomials are therefore continuous everywhere on the real line; rational functions are continuous on their domains. Piecewise definitions need a separate joining check. Equal-looking formulas on each open piece do not establish agreement at a boundary. A jump, a removable mismatch, an unbounded singularity and bounded oscillation are different failure mechanisms, even though all can defeat continuity at the joining point.

The intermediate value theorem states: if f is continuous on the closed real interval [a,b], every value between f(a) and f(b) is attained at some point in that interval. For a root, opposite signs suffice together with continuity. A concise existence proof uses bisection. If an endpoint or midpoint is a root, stop. Otherwise retain the half whose endpoints still have opposite signs. The nested closed intervals have lengths (b−a)/2ⁿ tending to zero. Their left endpoints increase and right endpoints decrease, so completeness gives a common limit c in every interval. Continuity sends the endpoint function values to f(c). Since those values remain on opposite sides of zero, the shared limit must be zero. This proves existence, not uniqueness; several crossings can occur.

Continuity cannot be omitted from that root certificate. The jump h(x)−1/2 changes sign across zero while taking only −1/2 and 1/2. It has no root. Closed endpoints also matter for attainment questions: f(x)=x on (0,1) approaches both bounds but attains neither. The extreme value theorem states that a continuous real function on a nonempty closed bounded interval attains a maximum and minimum. This is stronger than merely being bounded. For example 1/x on (0,1] is continuous on its domain but unbounded because the excluded endpoint can be approached.

Here is the mechanism behind the extreme value theorem. Every sequence in a closed bounded interval has a convergent subsequence whose limit remains in the interval: repeatedly bisect and retain a half containing infinitely many sequence terms, then select increasing indices from the nested halves. If continuous f were unbounded, choose x_n with |f(x_n)|>n; a convergent subsequence and continuity would force these outputs to converge to the finite value at its limit, a contradiction. Once f is bounded, let M be its output supremum and choose inputs with f(x_n)>M−1/n. A convergent subsequence with limit c then gives f(c)=M. Apply the same argument to −f for the minimum. The selection step can allow finitely many early exceptions; the increasing selected indices still force the errors to vanish.

This compactness reasoning extends to closed bounded sets in finite-dimensional real space by selecting subsequences successively for each coordinate. Thus the continuous quadratic energy on the unit sphere used in Module13 does attain its maximum. Closure ensures the limiting point remains on the sphere, boundedness permits subsequences, and continuity transfers output values. Infinite-dimensional spaces need additional care; this argument depends on finitely many coordinates.

A derivative, introduced fully next module, is a finite limit of difference quotients. If that quotient tends to D at a, it is bounded near a, and f(a+h)−f(a)=h times the quotient tends to zero. Hence differentiability implies continuity. The converse fails: |x| is continuous at zero but its left and right difference quotients are −1 and one. This boundary matters for AI objectives using absolute errors, maxima or threshold decisions. First establish which regularity property the proposed argument actually needs.

A series Σa_k is defined through its finite partial sums, not through an instruction to add infinitely many terms in a computer loop. For indexing from k=0, set S_N=Σ_{k=0}^{N−1}a_k, so N counts terms. The series converges to S when S_N→S. Terms and partial sums are different sequences. If partial sums converge, their differences a_N=S_{N+1}−S_N tend to zero. Thus terms tending to zero are necessary. They are not sufficient, as the harmonic series shows. A convergence test must certify partial sums, rather than observe small recent terms.

For a_k=Cq^k with q≠1, multiplying the finite sum by q and subtracting gives S_N=C(1−q^N)/(1−q). If |q|<1, q^N→0 and the sum is C/(1−q). The omitted tail is Cq^N/(1−q), and its magnitude is at most |C||q|^N/(1−|q|). For negative q the exact denominator |1−q| can give a tighter bound; the displayed bound handles both signs uniformly. If C≠0 and |q|≥1, the terms fail to approach zero, so the series diverges. C=0 is an exception: every term and partial sum is zero regardless of q. Always keep a boundary case rather than turning a sufficient pattern into a false universal claim.

The harmonic terms 1/n tend to zero, but grouping terms from n=2^{j−1}+1 through2^j gives 2^{j−1} terms, each at least1/2^j. Each block therefore contributes at least1/2. After any number of complete blocks the partial sum has increased by at least half that number, so the sums are unbounded. The infinite process cannot converge to a finite number. The proof gives a lower bound on arbitrarily distant sums; no finite screenshot can replace that quantifier.

For nonnegative terms, partial sums are increasing. The monotone convergence theorem from Section1 says they converge precisely when bounded above. This yields the comparison test: if 0≤a_n≤b_n eventually and Σb_n converges, Σa_n converges; finitely many initial terms only change its value by a finite amount. Conversely, a nonnegative lower comparison to a divergent series proves divergence. For n≥2, 1/n²≤1/[n(n−1)]=1/(n−1)−1/n. The comparison’s finite sum telescopes to1−1/N and is bounded by one. Including the first term bounds Σ1/n² by two, proving convergence without calculating its exact value.

Absolute convergence means Σ|a_n| converges. It implies convergence of Σa_n: write a_n=p_n−m_n with p_n=max(a_n,0), m_n=max(−a_n,0). Both nonnegative partial-sum sequences are bounded by the absolute-sum bound and therefore converge. Subtract their limits. This establishes the implication using completeness already taught. The converse can fail; the alternating harmonic series is a later optional proof. Rearranging finitely many terms is harmless, while rearranging an infinite conditionally convergent series needs a separate theorem. Do not transfer finite commutativity into an unjustified interchange of infinite limits.

The ratio test is another comparison argument. Suppose terms are eventually nonzero and |a_{n+1}/a_n|→L. If L<1, select q with L<q<1; beyond some N the ratios are at most q. Then |a_{N+j}|≤|a_N|q^j by induction, so a geometric comparison gives absolute convergence. If L>1, select a factor above one but below L; eventually magnitudes increase and cannot tend to zero. When L=1 the test gives no conclusion: both1/n and1/n² have ratios tending to one, yet their series behave differently. If infinitely many terms are zero, division-based ratios may be undefined; use another bound or analyse the nonzero pattern rather than ignore the domain issue.

For computation, a proven tail bound provides an error certificate for a finite approximation to a mathematically defined sum. Floating-point accumulation adds another source of error. The total discrepancy is bounded by truncation error plus arithmetic error, when both are bounded under an appropriate model. A tiny next term is a valid stopping certificate only when a theorem relates it to the remaining tail. For positive geometric q it does through1/(1−q); for harmonic terms it cannot, because the remaining sum is unbounded. Algorithms must attach stopping rules to the structure of the actual series.

An iterative method generates x_{k+1}=F(x_k). Convergence of this sequence is one question; correctness of its limit for the intended problem is another. If F is continuous and x_k→x*, passing to the limit in the recurrence gives x*=F(x*). Without continuity this passage needs justification, and without convergence the recurrence alone supplies no solution. A fixed point can also exist while a chosen starting point fails to approach it. The identity map has every point fixed, whereas F(x)=2x has fixed point zero but drives any nonzero initial value away from it.

A map is a contraction with factor q∈[0,1) on a set when |F(x)−F(y)|≤q|x−y| for every pair there. Suppose F maps a nonempty closed interval into itself and is such a contraction. Successive differences satisfy |x_{k+1}−x_k|≤q^k|x_1−x_0|. For m>n, the triangle inequality and finite geometric sum bound give |x_m−x_n|≤q^n|x_1−x_0|/(1−q). The iterates are Cauchy, hence converge in the reals; closure keeps their limit in the interval. A contraction is continuous by its inequality, so the limit is a fixed point. If two fixed points existed, their distance would be at most q times itself, forcing distance zero. This proves existence, convergence and uniqueness under the stated hypotheses. An unbounded closed interval also works, since the Cauchy bound supplies the needed boundedness.

Once a fixed point x* and a valid contraction factor are known, the current residual r=|F(x)−x| bounds error without knowing x*. Triangle inequality gives |x−x*|≤|x−F(x)|+|F(x)−F(x*)|≤r+q|x−x*|, hence |x−x*|≤r/(1−q). A computed rounded residual needs an arithmetic-error allowance before this becomes a rigorous numerical certificate. A contraction factor estimated from a few samples is also not a uniform theorem. State where the bound holds, whether all iterates stay there, and how the numerical residual was obtained.

Consider F(x)=x+.1(100−x), a contraction with q=.9 and fixed point100. Replace the update by x_new=x+α(F(x)−x), with α=10⁻¹⁰. At x=0 the applied step is10⁻⁹, below a tolerance10⁻⁶, while the error is almost100. The original residual is almost ten; its certified error bound is residual/(1−.9), almost100. Relaxation scales the step, so a tolerance on the applied update has a different meaning. Even when the relaxed map is a contraction, its factor is1−.1α, making its residual-to-error multiplier enormous. The same mathematics explains the failure rather than contradicting it.

Absolute and relative tolerances can be useful engineering choices: compare a discrepancy with atol+rtol·scale for a specified finite nonnegative scale. They encode an accepted approximation, not exact equality. Near zero the absolute term dominates; changing units changes the necessary absolute scale. Check residuals of the original equation, iterate movement, feasibility and any available error bounds according to the problem. A maximum iteration count can terminate an attempt but cannot certify convergence. Report which condition triggered termination so a downstream user can interpret the result.

Finite samples also cannot establish an infinite tail property. Lab1 constructs three sequences with the same first100 entries and different limiting behaviour. Lab3 samples a continuous narrow bump at10,001 regular grid points; every sampled height is zero, yet the height at its centre is one. This defeats a claim about that sampling grid, not continuity itself. Without a regularity bound controlling unsampled regions, a plot cannot certify a uniform error. A Lipschitz bound together with a maximum mesh spacing could supply such control, but obtaining the bound is additional mathematical evidence.

Finally, arithmetic can cause apparent stagnation: near10¹⁶ a binary64 computation may satisfy x+1==x because the intended increment rounds away. The real-number recurrence x_{k+1}=x_k+1 diverges, despite the stored value becoming unchanged. Inspect actual representable inputs as well as printed digits. Near a removable singularity, a nominal increment can round to zero and invoke a different domain case. Exact transformations, error bounds and finite observations play complementary roles; none should silently replace another in an AI training or numerical algorithm claim.

Lab 1 · Samples, exact sums and different tails