Continuity and the Intermediate Value Theorem
Define continuity at a point using three conditions: the function is defined, the limit exists, and they agree. Classify types of discontinuity. State and apply the Intermediate Value Theorem to locate roots.
A lesson of Pre-Calculus, in Limits: An Introduction to Calculus, about 10 minutes.
By the end of this lesson, you will be able to define continuity at a point using limits, classify discontinuities, determine intervals of continuity, and apply the Intermediate Value Theorem.
The Big Idea
Continuity is the formalization of "you can draw the graph without lifting your pencil." It requires three conditions — and all three must hold. Most functions are continuous on their domains; the interesting questions are about the exceptions (discontinuities) and what they reveal about the function.
Definition of Continuity at a Point
$f$ is continuous at $x = c$ iff all three hold:
- $f(c)$ is defined
- $\lim_{x \to c} f(x)$ exists
- $\lim_{x \to c} f(x) = f(c)$
If any condition fails, $f$ has a discontinuity at $c$.
Types of Discontinuity
Removable (hole): the limit exists, but $f(c) \neq L$ or is undefined. The "hole" can be "filled" by redefining $f(c) = L$.
Jump: one-sided limits exist but are not equal ($L^- \neq L^+$). Cannot be removed.
Infinite: $f \to \pm\infty$ as $x \to c$ (vertical asymptote). Cannot be removed.
Oscillatory: $f$ oscillates near $c$ without settling (e.g., $\sin(1/x)$ at $x = 0$). Cannot be removed.
Example 1 — Checking Continuity
$$f(x) = \frac{x^2 - 1}{x - 1}$$
At $x = 1$: $f(1)$ is undefined → discontinuous at $x = 1$.
$\lim_{x \to 1} f(x) = \lim_{x \to 1}(x+1) = 2$ (removable discontinuity).
If we redefine $f(1) = 2$, it becomes continuous.
Example 2 — Piecewise Continuity
$$g(x) = \begin{cases} x + 3 & x < 2 \ 7 & x = 2 \ 2x + 2 & x > 2 \end{cases}$$
At $x = 2$:
- $g(2) = 7$ ✓ (defined)
- $\lim_{x \to 2^-} = 5$, $\lim_{x \to 2^+} = 6$ → limit DNE ✗
Jump discontinuity at $x = 2$.
Continuity on an Interval
$f$ is continuous on $(a, b)$ if continuous at every point in $(a,b)$.
$f$ is continuous on $[a, b]$ if: continuous on $(a,b)$, $\lim_{x \to a^+}f(x) = f(a)$, and $\lim_{x \to b^-}f(x) = f(b)$.
Standard continuous functions: polynomials (all reals), rationals (where denominator ≠ 0), trig functions (on their domains), exponential and log functions (on their domains), roots.