Continuity

A function is continuous at a point if its graph passes through that point without lifting the pen. Formally, continuity links a function's value at $a$ to the limiting behaviour of $f(x)$ as $x \to a$. The chapter's first task is to translate this intuitive picture into testable conditions, then identify the kinds of failures that can occur.

Definition

A function $f$ is continuous at the point $a$ in its domain if

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

Three conditions must hold:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists (i.e., left and right limits agree).
3. The two are equal.

If $f$ is continuous at every point of an interval, we say $f$ is continuous on that interval.

One-sided continuity

$f$ is left-continuous at $a$ if $\lim_{x \to a^-} f(x) = f(a)$. It is right-continuous if $\lim_{x \to a^+} f(x) = f(a)$. Two-sided continuity at $a$ requires both.

For points at the boundary of the domain (e.g., $a = 0$ for $f : [0, \infty) \to \mathbb{R}$), only one-sided continuity is meaningful.

Types of discontinuity

When $f$ fails to be continuous at $a$, we classify the failure:

• Removable discontinuity. $\lim_{x \to a} f(x)$ exists but does not equal $f(a)$ (or $f(a)$ is undefined). Redefining $f(a)$ fixes the problem.
• Jump discontinuity. Left and right limits both exist but differ.
• Infinite discontinuity. A one-sided limit is $\pm \infty$.
• Oscillatory discontinuity. The limit fails to exist due to oscillation (e.g., $\sin(1/x)$ at $0$).

Algebra of continuous functions

If $f$ and $g$ are continuous at $a$, then so are $f + g, f - g, fg$, and $f/g$ (the last provided $g(a) \neq 0$). Composition is continuous: if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f \circ g$ is continuous at $a$.

These rules let us conclude: every polynomial is continuous on $\mathbb{R}$; every rational function is continuous wherever the denominator is nonzero; $\sin, \cos, e^x$ are continuous on $\mathbb{R}$; $\tan, \sec$ are continuous wherever they are defined.

Important special cases

$f(x) = \dfrac{\sin x}{x}$ at $x = 0$: limit is $1$, but $f(0)$ is undefined. Removable.

$f(x) = \begin{cases} 1, & x \ge 0 \\ -1, & x < 0 \end{cases}$ at $x = 0$: left limit $-1$, right limit $1$. Jump.

$f(x) = 1/x$ at $x = 0$: limit is $\pm \infty$. Infinite.

$f(x) = \sin(1/x)$ at $x = 0$: limit does not exist. Oscillatory.

Continuity on an interval

$f$ is continuous on $[a, b]$ if continuous at every interior point and right-continuous at $a$, left-continuous at $b$. Boundedness and attainment of extrema are guaranteed by continuity on a closed interval (Extreme Value Theorem), as is the intermediate value property.

Worked examples

Example 1. Check continuity of $f(x) = \begin{cases} x^2, & x \le 1 \\ 2x - 1, & x > 1 \end{cases}$ at $x = 1$.

Left limit: $\lim_{x \to 1^-} x^2 = 1$. Right limit: $\lim_{x \to 1^+}(2x - 1) = 1$. $f(1) = 1$. All three agree. Continuous.

Example 2. Check continuity of $f(x) = \dfrac{x^2 - 1}{x - 1}$ at $x = 1$.

Undefined at $x = 1$, but $\lim_{x \to 1} \dfrac{x^2 - 1}{x - 1} = \lim_{x \to 1}(x + 1) = 2$. Removable discontinuity; can be made continuous by defining $f(1) = 2$.

Example 3. Find $k$ so that $f(x) = \begin{cases} kx + 1, & x \le 2 \\ x^2 - 3, & x > 2 \end{cases}$ is continuous at $x = 2$.

Left value $f(2) = 2k + 1$. Right limit $\lim_{x \to 2^+}(x^2 - 3) = 1$. Equate: $2k + 1 = 1$, so $k = 0$.

Example 4. Show $f(x) = |x|$ is continuous at $x = 0$.

Left limit: $\lim_{x \to 0^-}|x| = 0$. Right limit: $\lim_{x \to 0^+}|x| = 0$. $f(0) = 0$. Continuous.

Example 5. Determine continuity of $f(x) = \begin{cases} \dfrac{\sin x}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ at $x = 0$.

Limit $1$, value $0$. Not continuous; this is a removable discontinuity.

Example 6. Show $f(x) = \lfloor x \rfloor$ (greatest integer) is discontinuous at every integer.

At $n \in \mathbb{Z}$: left limit $\lim_{x \to n^-}\lfloor x \rfloor = n - 1$; right limit $\lim_{x \to n^+}\lfloor x \rfloor = n$. Jump of $1$.

Try it yourself

1. Check continuity of $f(x) = \sin x \cdot \cos x$ on $\mathbb{R}$.
2. At $x = 0$, is $f(x) = x^2$ continuous?
3. Is $f(x) = 1/x$ continuous at $x = 0$?
4. Determine $k$ so that $f(x) = \begin{cases} k \sin x, & x \le 0 \\ x + 2, & x > 0 \end{cases}$ is continuous at $x = 0$.
5. Classify the discontinuity of $f(x) = \dfrac{x^2 - 4}{x - 2}$ at $x = 2$.
6. Show: every polynomial is continuous on $\mathbb{R}$.
7. Show: every constant function is continuous.
8. Is $f(x) = \dfrac{|x|}{x}$ continuous at $x = 0$?
9. Show: if $f$ is continuous on $[a, b]$ then $|f|$ is continuous there.
10. Find the points where $f(x) = \tan x$ is discontinuous.
11. Examine continuity of $f(x) = \dfrac{x^3 - 1}{x - 1}$ at $x = 1$.
12. Find $a, b$ such that $f(x) = \begin{cases} ax + 1, & x \le 1 \\ b x + 2, & x > 1 \end{cases}$ is continuous and differentiable at $x = 1$.
13. Investigate continuity of $f(x) = x \sin(1/x)$ at $x = 0$.
14. Use the squeeze theorem to show $\lim_{x \to 0} x^2 \sin(1/x) = 0$.

Pitfalls / Tricks

• Continuity at $a$ requires the function to be defined at $a$. A "limit equals value" check needs both.
• One-sided limits matter: a function with different left and right limits is discontinuous, even if both exist.
• Removable discontinuities can be "fixed" by redefining the function at one point.
• The greatest integer function $\lfloor x \rfloor$ is discontinuous exactly at integers.
• $\sin(1/x)$ oscillates near $0$ , limit doesn't exist; multiply by $x$ to tame it.

Next we ask when a continuous function also has a tangent line.