Limits and Continuity of Real-Valued Functions

⚡ Key Takeaways

Limits describe the value a function approaches as its input approaches a point; continuity demands that this limiting value actually equals the function's value there. One-line: $\lim_{x\to a} f(x) = L$ means $f(x)$ can be forced arbitrarily close to $L$ by taking $x$ close enough to $a$ (but not equal). Applies wherever functions on $\mathbb{R}$ must be analyzed for smoothness, approximation, optimization, and convergence. Formalized rigorously by Cauchy, Bolzano, and Weierstrass in the 19th century.

💡 The Core Insight

A limit is a statement about a challenge-response game: an adversary names a tolerance $\varepsilon > 0$ around the candidate value $L$ ; you must produce a neighborhood radius $\delta > 0$ around $a$ guaranteeing every input within $\delta$ (excluding $a$ itself) lands within $\varepsilon$ of $L$ . The value $f(a)$ — and even whether it exists — is irrelevant to the limit. Continuity is precisely the demand that you can win this game with $L = f(a)$ , eliminating the gap between the approach and the destination.

DefinitionLimit of a Function (ε–δ)

Let $f$ be defined on a punctured neighborhood of $a \in \mathbb{R}$ . We say $\lim_{x\to a} f(x) = L$ if:

1

\quad \forall \varepsilon > 0 \;\; \exists \delta > 0 \;\; \forall x \;\; \big(0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon\big)

Derivation hint: Read inner-to-outer — given the tolerance $\varepsilon$ , the existential $\delta$ promises a punctured interval $(a-\delta, a+\delta)\setminus\{a\}$ mapping into $(L-\varepsilon, L+\varepsilon)$ . Here $\varepsilon$ is the output tolerance, $\delta$ the input radius, $a$ the limit point, $L$ the limit value, and $|x-a|$ , $|f(x)-L|$ are Euclidean distances on $\mathbb{R}$ . The condition $0 < |x-a|$ explicitly excludes $x = a$ .

DefinitionOne-Sided Limits

The right-hand and left-hand limits are:

2

\quad \lim_{x\to a^+} f(x) = L^+ \iff \forall\varepsilon>0\;\exists\delta>0\;\;(a < x < a+\delta \implies |f(x)-L^+|<\varepsilon)

Derivation hint: Restrict the punctured neighborhood to its right half $a<x<a+\delta$ ; the left-hand version uses $a-\delta<x<a$ . The two-sided limit exists iff both one-sided limits exist and are equal:

3

\quad \lim_{x\to a} f(x) = L \iff \lim_{x\to a^+} f(x) = \lim_{x\to a^-} f(x) = L

Derivation hint: Splitting the punctured neighborhood into left and right halves shows the two-sided $\delta$ is the minimum of the one-sided $\delta$ 's.

DefinitionContinuity at a Point

$f$ is continuous at $a$ if $a \in \mathrm{dom}(f)$ and:

4

\quad \lim_{x\to a} f(x) = f(a)

Derivation hint: This single equation silently asserts three conditions — $f(a)$ exists, the limit exists, and they are equal. Equivalently, in ε–δ form (note $0 < |x-a|$ becomes just $|x-a|$ , since $x=a$ trivially satisfies the conclusion):

5

\quad \forall\varepsilon>0\;\exists\delta>0\;\;(|x-a|<\delta \implies |f(x)-f(a)|<\varepsilon)

Derivation hint: Set $L = f(a)$ in [1]; the point $x=a$ may be included because $|f(a)-f(a)| = 0 < \varepsilon$ holds automatically.

DefinitionContinuity on a Set

$f$ is continuous on $S \subseteq \mathbb{R}$ if it is continuous at every point of $S$ (using one-sided continuity at endpoints). $f$ is uniformly continuous on $S$ if:

6

\quad \forall\varepsilon>0\;\exists\delta>0\;\forall x,y\in S\;\;(|x-y|<\delta \implies |f(x)-f(y)|<\varepsilon)

Derivation hint: Compare with [5] — here a single $\delta$ must work for <strong>all</strong> points simultaneously; the quantifier $\forall x$ moved inside, after $\exists\delta$ .

TheoremSequential Characterization (Heine)

$\lim_{x\to a} f(x) = L$ if and only if for every sequence $(x_n)$ with $x_n \to a$ and $x_n \neq a$ :

7

\quad \lim_{n\to\infty} f(x_n) = L

Derivation hint: Forward direction uses ε–δ on the tail of $(x_n)$ ; reverse direction is by contraposition, building a violating sequence $x_n$ with $|x_n-a|<1/n$ but $|f(x_n)-L|\ge\varepsilon_0$ . This converts function-limit problems into sequence-limit problems, and is the standard tool for disproving limits (find two sequences giving different values).

TheoremAlgebraic Limit Laws

If $\lim_{x\to a} f = L$ and $\lim_{x\to a} g = M$ , then:

8

\quad \lim_{x\to a}(f+g) = L+M,\quad \lim_{x\to a}(fg) = LM,\quad \lim_{x\to a}\frac{f}{g} = \frac{L}{M}\ (M\neq 0)

Derivation hint: For the sum, $|(f+g)-(L+M)| \le |f-L|+|g-M|$ ; force each below $\varepsilon/2$ . For products, add and subtract $Lg(x)$ .

TheoremSqueeze Theorem

If $g(x) \le f(x) \le h(x)$ near $a$ and $\lim_{x\to a} g = \lim_{x\to a} h = L$ , then:

9

\quad \lim_{x\to a} f(x) = L

Derivation hint: Both $g$ and $h$ lie in $(L-\varepsilon, L+\varepsilon)$ ; the squeezed $f$ is trapped between them, hence also in that interval.

ProofThe Limit of a Composition / Continuity is Preserved

We prove that the composition of continuous functions is continuous: if $g$ is continuous at $a$ and $f$ is continuous at $b = g(a)$ , then $f\circ g$ is continuous at $a$ .

Step 1. Fix $\varepsilon > 0$ . We must find $\delta>0$ so that $|x-a|<\delta \implies |f(g(x)) - f(g(a))| < \varepsilon$ .

Step 2. Since $f$ is continuous at $b = g(a)$ , apply [5] at $b$ : there exists $\eta > 0$ such that

10

\quad |y - b| < \eta \implies |f(y) - f(b)| < \varepsilon.

Derivation hint: This is literally continuity of $f$ at $b$ with tolerance $\varepsilon$ producing inner radius $\eta$ .

Step 3. Since $g$ is continuous at $a$ , apply [5] at $a$ with tolerance $\eta$ : there exists $\delta > 0$ such that

11

\quad |x - a| < \delta \implies |g(x) - g(a)| < \eta.

Derivation hint: Feed the radius $\eta$ from Step 2 as the output tolerance required of $g$ .

∎

ProofContinuity ⟹ Boundedness on Compact Intervals (Extreme Value Theorem, sketch)

Claim. A continuous $f:[a,b]\to\mathbb{R}$ attains a maximum and minimum.

Step 1. Boundedness. Suppose not bounded above. Then for each $n$ pick $x_n\in[a,b]$ with $f(x_n) > n$ . By Bolzano–Weierstrass, $(x_n)$ has a convergent subsequence $x_{n_k}\to c\in[a,b]$ .

Step 2. By sequential continuity [7], $f(x_{n_k}) \to f(c)$ , a finite number. But $f(x_{n_k}) > n_k \to \infty$ , contradiction. Hence $f$ is bounded.

Step 3. Attainment. Let $M = \sup f([a,b])$ . Choose $y_n\in[a,b]$ with $f(y_n) \to M$ . A convergent subsequence $y_{n_k}\to d$ gives $f(d) = \lim f(y_{n_k}) = M$ . So the supremum is achieved. The minimum follows by applying the result to $-f$ . $\blacksquare$

∎

AC

Augustin-Louis Cauchy

1789–1857

Cauchy's Cours d'analyse (1821) was the first systematic attempt to ground calculus on limits rather than infinitesimals, defining a limit as a value that variables "approach indefinitely so as to differ from it by as little as one wishes." Though his phrasing was still verbal, it contained the seed of the ε–δ method. Cauchy famously (and incorrectly) "proved" that a convergent sum of continuous functions is continuous — an error later corrected by uniform convergence — illustrating how subtle limiting processes are. His contemporary Bernard Bolzano had reached similar rigor independently a few years earlier (1817), but his work went largely unnoticed. The fully symbolic ε–δ formulation we use today crystallized in the lectures of Karl Weierstrass in the 1860s.

🧠 Intuition

Think of $f(x)$ as a machine and $L$ as a target on a dartboard with concentric rings. The adversary shrinks the bullseye to radius $\varepsilon$ . You respond by restricting your throwing zone to inputs within $\delta$ of $a$ . You win the limit if there is always a throwing zone small enough that every dart lands inside the current bullseye. Crucially, you remove the single point $x=a$ from the throwing zone — the limit doesn't care what happens at $a$ , only what happens near it. Continuity is the stricter demand that the bullseye center sits exactly at $f(a)$ : the function's actual value at $a$ is where everything is headed, so there's no jump, no hole, no surprise.

Figure

The ε–δ game: inputs within δ map into the ε-band around L

ExampleProving a Polynomial Limit by ε–δ

Given: Prove $\lim_{x\to 3}(2x + 1) = 7$ .

Step 1 — Scratch work. We need $|(2x+1) - 7| < \varepsilon$ , i.e. $|2x - 6| = 2|x-3| < \varepsilon$ .

Step 2 — Solve for $|x-3|$ . This requires $|x-3| < \varepsilon/2$ . So choose $\delta = \varepsilon/2$ .

Step 3 — Formal proof. Let $\varepsilon > 0$ . Set $\delta = \varepsilon/2$ . Suppose $0 < |x-3| < \delta$ . Then

12

\quad |(2x+1)-7| = 2|x-3| < 2\delta = 2\cdot\frac{\varepsilon}{2} = \varepsilon.

Derivation hint: Factor the constant 2 out of the absolute value, then substitute the chosen $\delta$ .

Answer: $\boxed{\lim_{x\to 3}(2x+1) = 7,\ \text{with } \delta = \varepsilon/2}$ (dimensionless).

ExampleA Removable Discontinuity and a Quadratic ε–δ

Given: Evaluate $\lim_{x\to 2}\dfrac{x^2 - 4}{x - 2}$ and verify by ε–δ that the limit is $4$ .

Step 1 — Simplify. For $x\neq 2$ : $\dfrac{x^2-4}{x-2} = \dfrac{(x-2)(x+2)}{x-2} = x+2$ . So the limit is $2+2 = 4$ . The function has a removable discontinuity (hole) at $x=2$ .

Step 2 — ε–δ for $x+2$ . Need $|(x+2)-4| = |x-2| < \varepsilon$ . Choose $\delta = \varepsilon$ .

Step 3 — Verify. Let $\varepsilon>0$ , set $\delta=\varepsilon$ . If $0<|x-2|<\delta$ then (with $x\neq 2$ , so the cancellation is valid):

13

\quad \left|\frac{x^2-4}{x-2} - 4\right| = |(x+2)-4| = |x-2| < \delta = \varepsilon.

Derivation hint: The condition $0<|x-2|$ guarantees $x\neq 2$ , legitimizing the factor cancellation before applying $\delta$ .

Answer: $\boxed{\lim_{x\to 2}\dfrac{x^2-4}{x-2} = 4}$ . The function $\frac{x^2-4}{x-2}$ becomes continuous if we redefine its value at $2$ to be $4$ .

⚠️ Common mistake

"The limit equals the function value." False in general — the limit ignores $x=a$ entirely. A limit can exist where $f(a)$ is undefined (Example 2), and $f(a)$ can exist while differing from the limit (a removable discontinuity with a misplaced point). The equality $\lim_{x\to a} f = f(a)$ is precisely the definition of continuity, not an automatic fact.

"Continuous means differentiable." False. The Weierstrass function is continuous everywhere but differentiable nowhere.

"A single $\delta$ works for all points." That stronger property is uniform continuity [6], which is not implied by ordinary continuity on non-compact sets (see counterexample).

CounterexampleContinuous but not uniformly continuous

Let $f(x) = 1/x$ on $(0,1]$ . It is continuous at every point of $(0,1]$ . But it is not uniformly continuous: take $x_n = \frac{1}{n}$ and $y_n = \frac{1}{n+1}$ . Then $|x_n - y_n| = \frac{1}{n(n+1)} \to 0$ , yet

14

\quad |f(x_n)-f(y_n)| = |n - (n+1)| = 1 \not\to 0.

Derivation hint: As $x\to 0^+$ the graph steepens without bound, so no single $\delta$ can control the output gap everywhere; the required $\delta$ shrinks toward $0$ . No single $\delta$ can satisfy [6] for, say, $\varepsilon = \tfrac12$ . (On the compact interval $[\,c,1]$ with $c>0$ , the Heine–Cantor theorem guarantees uniform continuity.)

CounterexampleA limit that fails to exist by oscillation

$f(x) = \sin(1/x)$ has no limit as $x\to 0$ . Using [7], take $x_n = \frac{1}{n\pi} \to 0$ giving $f(x_n) = 0$ , but $z_n = \frac{1}{(2n+\frac12)\pi}\to 0$ giving $f(z_n) = 1$ . Two sequences, two distinct sub-limits $\Rightarrow$ the limit cannot exist. (By contrast $x\sin(1/x)\to 0$ by the Squeeze Theorem [9], since $|x\sin(1/x)|\le|x|$ .)

connection

Limits are the foundation on which the entire edifice of calculus rests. The Derivative is defined as $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ , a limit of a difference quotient; differentiability is therefore a strengthening of continuity. The Riemann Integral is a limit of Riemann sums as the mesh of partitions tends to zero. The convergence of Infinite Series and Sequences of Real Numbers uses the same ε–N machinery, related to ε–δ by the sequential characterization [7]. Continuity generalizes to Metric Spaces and Topological Spaces, where the open-set definition ("preimages of open sets are open") reveals the topological essence that ε–δ merely encodes for $\mathbb{R}$ . The Intermediate Value Theorem and Extreme Value Theorem are the two great existence theorems flowing directly from continuity on compact connected sets.

🔬 Where this appears

⏳ History

- c. 1670s — Newton and Leibniz develop calculus using "fluxions" and infinitesimals, with limits described intuitively and without rigor.

- 1734 — Bishop George Berkeley's The Analyst attacks the logical incoherence of infinitesimals ("ghosts of departed quantities"), spurring the search for rigor.

- 1817 — Bernard Bolzano gives a purely analytic proof of the Intermediate Value Theorem and an early correct definition of continuity, anticipating ε–δ — but his work is overlooked for decades.

- 1821 — Cauchy's Cours d'analyse founds analysis on limits, defining continuity and convergence verbally, though still imprecisely (mixing pointwise and uniform notions).

- 1854 — Riemann formalizes the integral as a limit of sums in his Habilitation, requiring a sharp limit concept.

- c. 1861–1872 — Karl Weierstrass, in Berlin lectures, introduces the fully symbolic ε–δ definition and constructs his everywhere-continuous, nowhere-differentiable function (published 1872), shattering geometric intuition.

- 1872 — Heine and Cantor establish uniform continuity and the Heine–Cantor theorem (continuity on a compact set ⟹ uniform continuity); Cantor and Dedekind rigorously construct

\mathbb{R}

via Cauchy sequences / cuts.

- 1875 — Darboux clarifies the relationship between Riemann integrability and continuity.

- 1906 — Fréchet introduces metric spaces, generalizing limits and continuity beyond

\mathbb{R}

and seeding modern topology.

open-problem

The ε–δ framework, while complete for $\mathbb{R}$ , leaves open foundational and computational questions. Constructive analysis (Bishop, Brouwer) rejects the law of excluded middle, so the classical IVT — which uses non-constructive bisection limit arguments — fails to yield a computable root in general; characterizing exactly which continuous functions admit constructive root-finding remains active. In descriptive set theory, the structure of discontinuity sets is fully classified (the set of discontinuities of any function is an $F_\sigma$ set, and conversely every $F_\sigma$ set arises this way — Young's theorem), but analogous questions for higher-order regularity (sets of non-differentiability, sets where derivatives fail to be continuous) involve deep questions about Baire classes still under investigation. Finally, in nonstandard analysis (Robinson, 1960s), limits and continuity are rephrased using genuine infinitesimals from a hyperreal field $^*\mathbb{R}$ — continuity at $a$ becomes " $x\approx a \implies f(x)\approx f(a)$ " for hyperreals — raising ongoing pedagogical and philosophical debate about which formulation is "primary."

Limits and Continuity of Real-Valued Functions

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also