The Timpson Lyceum · Calculus 1 (2024)

Continuous Functions & Continuity at a Point - Lecture 6 (Calculus 1)

Overview

This lecture introduces continuity, the second of calculus's four big ideas (limits, continuity, derivatives, integrals). Building directly on the precise definition of a limit, it defines what it means for a function to be continuous at a point, then extends that idea to endpoints of a closed interval and to whole functions. It catalogs the common types of discontinuity (removable, jump, infinite, oscillating), shows that the usual arithmetic and composition rules for limits carry over to continuity, and finishes with two important applications: the Intermediate Value Theorem and continuous extensions. Because nearly every function we work with is continuous except at a handful of strange points, the goal is to recognize and reason about those exceptional points.

Key Concepts

Continuity at a point (simple form): A function $f$ is continuous at $x = c$ if and only if $\lim_{x \to c} f(x) = f(c).$ Three things must all hold: $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and the two are equal. This is just the limit definition with the target value $L$ replaced by the function's actual value $f(c)$.

Epsilon–delta form: $f$ is continuous at $c$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that whenever $|x - c| < \delta$, then $|f(x) - f(c)| < \varepsilon$. Identical to the limit definition except $L$ becomes $f(c)$, so the function approaches its own value at the point.

Why a limit can fail to equal the value:

• If the function jumps (different left and right behavior), the two-sided limit does not exist, so it cannot be continuous there.
• If there is a hole with the value placed elsewhere, the limit exists but does not equal $f(c)$, so it is not continuous there either. (Limits ignore what happens at $c$; continuity does not.)

Continuity at endpoints: For a function defined on a closed interval $[a, b]$, one-sided limits are used at the ends:

• Continuous at the left endpoint $a$ if $\displaystyle\lim_{x \to a^+} f(x) = f(a)$.
• Continuous at the right endpoint $b$ if $\displaystyle\lim_{x \to b^-} f(x) = f(b)$.

Continuous on its domain: A function is continuous if it is continuous at every point in its domain. For example, $f(x) = 1/x$ is a continuous function even though it blows up at $0$ — because $0$ is not in its domain.

Types of discontinuity:

• Removable discontinuity — a single missing/misplaced point (a hole) that can be "plugged" to fix continuity. Example: $\dfrac{\sin x}{x}$ at $x = 0$.
• Jump (step/gap) discontinuity — the function jumps between values, as in the floor function $\lfloor x \rfloor$, which is continuous only on each open interval $(n, n+1)$.
• Infinite discontinuity — the function shoots off toward $\pm\infty$, as with $1/x$ at $0$.
• Oscillating discontinuity — the function oscillates infinitely fast and approaches no value, as with $\sin(1/x)$ near $0$.

Continuity laws (inherited from limit laws): If $f$ and $g$ are continuous at $c$, then so are:

• $f + g$, $f - g$, $f \cdot g$;
• $f/g$, provided $g(c) \neq 0$;
• $k f$ for a constant $k$, and $f^n$ for a positive integer $n$;
• $\sqrt[n]{f}$ for a positive integer $n$;
• the composition $g \circ f$, provided $g$ is continuous at $f(c)$.

Consequences: Every polynomial is continuous everywhere ($\lim_{x \to c} P(x) = P(c)$); every rational function $P(x)/Q(x)$ is continuous wherever $Q(c) \neq 0$; and the absolute value function is continuous on all of $\mathbb{R}$. Combined with composition, this shows expressions like $\sqrt{x^2 - 2x + 5}$ or $|x - 2|/(x^2 - 2)$ are continuous on their natural domains.

Limit-inside-composition theorem: If $g$ is continuous at $b$ and $\lim_{x \to c} f(x) = b$, then $\lim_{x \to c} g\big(f(x)\big) = g\Big(\lim_{x \to c} f(x)\Big) = g(b).$ The corollary that composition preserves continuity follows immediately when $b = f(c)$.

Intermediate Value Theorem (IVT): If $f$ is continuous on a closed interval $[a, b]$ and $y_0$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c$ in $[a, b]$ with $f(c) = y_0$. It guarantees a solution exists (at least one $c$) without locating it; commonly used to confirm an equation has a root before approximating it.

Continuous extension: Given a function with a removable hole, define a new function that agrees everywhere except at the bad point, where it takes the value of the limit. This "fills the hole," producing a function that is continuous there. In practice you either set the value equal to the limit at that point, or algebraically cancel the common factor causing the hole.

Worked Example

Use the IVT to show that $x^3 - x - 1 = 0$ has a solution. Let $f(x) = x^3 - x - 1$, which is a polynomial and therefore continuous on every closed interval. Evaluate the endpoints, then invoke the IVT with the target value $y_0 = 0$:

Since $f$ changes sign from $-1$ at $x = 1$ to $5$ at $x = 2$ and is continuous, the IVT guarantees some $c$ in $[1, 2]$ where $f(c) = 0$ — a root of the equation. (A numeric method could then narrow in on its value.)

As a second, shorter illustration, a continuous extension of $f(x) = \dfrac{x^2 + x - 6}{x^2 - 4}$ at $x = 2$ comes from factoring and canceling the offending factor:

so defining $F(x) = \dfrac{x + 3}{x + 2}$ fills the hole, and $F$ is continuous at $x = 2$.

Summary

• A function is continuous at $c$ exactly when $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and the two are equal; continuity is the limit definition with $L$ replaced by $f(c)$.
• At the endpoints of a closed interval, use the appropriate one-sided limit; a function is continuous overall when it is continuous at every point of its domain.
• The four common discontinuities are removable (a hole), jump/step, infinite, and oscillating.
• All limit laws carry over: sums, differences, products, quotients (nonzero denominator), powers, roots, and compositions of continuous functions are continuous; polynomials, rational functions (off their zeros), and $|x|$ are continuous.
• The Intermediate Value Theorem guarantees a continuous function on $[a, b]$ attains every value between $f(a)$ and $f(b)$ — useful for proving a solution exists.
• A continuous extension repairs a removable discontinuity by setting the function's value at the bad point equal to the limit there, often via algebraic cancellation.