Continuous Function

Limits tell you what a function approaches at a point. Continuity asks whether the function arrives — whether the limit equals the actual value. Intuitively, a function is continuous if small changes in input produce only small changes in output, with no sudden jumps or gaps.

Continuity at a point

Definition. A function $f : D \to \mathbb{R}$ is continuous at $a \in D$ if

$$\lim_{x \to a} f(x) = f(a).$$

This packages three conditions into one: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, and (3) the limit equals $f(a)$.

In ε–δ terms, $f$ is continuous at $a$ if and only if for every $\varepsilon > 0$ there exists $\delta > 0$ such that

$$x \in D \text{ and } |x - a| < \delta \implies |f(x) - f(a)| < \varepsilon.$$

Notice that unlike the limit definition, $x$ is now allowed to equal $a$ — the condition still holds trivially there.

If $a$ is an isolated point of $D$ — some neighborhood of $a$ contains no other point of $D$ — then $f$ is automatically continuous at $a$, because the limit condition is vacuously satisfied.

Continuity on a set

Definition. $f$ is continuous on $S \subseteq D$ if it is continuous at every point of $S$. When $S = D$, one says simply that $f$ is continuous.

For functions defined on a closed interval $[a, b]$, continuity at the endpoints is interpreted one-sidedly: $\lim_{x \to a^+} f(x) = f(a)$ and $\lim_{x \to b^-} f(x) = f(b)$.

Algebra of continuous functions

The arithmetic of limits carries over immediately.

Theorem. If $f$ and $g$ are continuous at $a$, then so are:

• $f + g$, $f - g$, $f \cdot g$
• $f / g$, provided $g(a) \neq 0$
• $c \cdot f$ for any constant $c$

Proof. For $f + g$: since $\lim_{x \to a} f(x) = f(a)$ and $\lim_{x \to a} g(x) = g(a)$, the limit sum rule gives $\lim_{x \to a}(f + g)(x) = f(a) + g(a) = (f + g)(a)$. The remaining cases follow analogously from the corresponding limit rules. $\square$

Composition

Theorem. If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f \circ g$ is continuous at $a$.

Proof. Let $\varepsilon > 0$. Continuity of $f$ at $g(a)$ gives $\eta > 0$ such that $|y - g(a)| < \eta$ implies $|f(y) - f(g(a))| < \varepsilon$. Continuity of $g$ at $a$ gives $\delta > 0$ such that $|x - a| < \delta$ implies $|g(x) - g(a)| < \eta$. Combining: $|x - a| < \delta$ implies $|f(g(x)) - f(g(a))| < \varepsilon$. $\square$

Elementary functions are continuous

Every elementary function is continuous on its natural domain.

Function	Continuous on
Polynomials	$\mathbb{R}$
Rational $p/q$	$\{x : q(x) \neq 0\}$
$e^x$	$\mathbb{R}$
$\ln x$	$(0, \infty)$
$\sin x$, $\cos x$	$\mathbb{R}$
$\tan x$	$\{x : \cos x \neq 0\}$

Polynomials are continuous because $\lim_{x \to a} x = a$ and limit arithmetic extends this to all monomials and then all polynomials. Each other elementary function is proved continuous using its specific definition or series expansion; compositions preserve continuity by the theorem above.

This means you can evaluate limits of elementary expressions simply by substituting the point, as long as the function is defined there.

Discontinuities

When $f$ fails to be continuous at $a$, the failure is classified by what does happen:

Type	Behavior
Removable	$\lim_{x \to a} f(x)$ exists but $\neq f(a)$, or $f(a)$ is undefined
Jump	Both one-sided limits exist but differ: $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$
Essential	At least one one-sided limit does not exist (or is $\pm\infty$)

A removable discontinuity can be fixed by redefining $f(a) \coloneqq \lim_{x \to a} f(x)$. Jump and essential discontinuities cannot be patched this way.

Summary

• $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$ — the limit exists and matches the function value.
• Continuous on a set means continuous at every point of that set.
• Sums, differences, products, quotients (nonzero denominator), and compositions of continuous functions are continuous.
• Every elementary function is continuous on its natural domain; limits of elementary expressions are computed by direct substitution.
• Discontinuities are removable, jump, or essential — only removable discontinuities can be fixed by redefining the function value.