2.4 Continuity

Continuity

Continuity describes functions whose graphs have no breaks, holes, or jumps. Understanding continuity is essential because many of the major theorems in calculus (including those behind derivatives and integrals) only work when functions are continuous. This section covers the formal definition, the types of discontinuities you'll encounter, and key tools like the Intermediate Value Theorem.

Continuity at Points and Intervals

A function is continuous at a point $a$ when three conditions are all satisfied:

1. $f(a)$ is defined (the function has an actual value at $a$)
2. $\lim_{x \to a} f(x)$ exists (the left-hand and right-hand limits agree)
3. $\lim_{x \to a} f(x) = f(a)$ (the limit equals the function value)

If any one of these fails, the function is discontinuous at $a$. When you're checking continuity, work through these three conditions in order. The third condition is the one students most often forget to verify.

Continuity on a closed interval $[a, b]$ means:

• The function is continuous at every point in the open interval $(a, b)$
• The function is right-continuous at the left endpoint: $\lim_{x \to a^+} f(x) = f(a)$
• The function is left-continuous at the right endpoint: $\lim_{x \to b^-} f(x) = f(b)$

At the endpoints, you can only approach from inside the interval, so you only need the one-sided limit from that direction.

Types of Discontinuities

Removable Discontinuity

A removable discontinuity occurs at $x = a$ when $\lim_{x \to a} f(x)$ exists, but either $f(a)$ is undefined or $f(a) \neq \lim_{x \to a} f(x)$. On a graph, this looks like a small hole in the curve.

It's called "removable" because you could fix it by redefining $f(a)$ to equal the limit. For example, $f(x) = \frac{x^2 - 1}{x - 1}$ is undefined at $x = 1$, but $\lim_{x \to 1} f(x) = 2$. Redefining $f(1) = 2$ removes the discontinuity.

Jump Discontinuity

A jump discontinuity occurs at $x = a$ when both one-sided limits exist but are not equal:

$\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$

The function literally "jumps" from one value to another. This is common in piecewise functions where different formulas apply on either side of a point. Jump discontinuities cannot be removed by redefining a single point.

Infinite Discontinuity

An infinite discontinuity occurs at $x = a$ when at least one of the one-sided limits is $+\infty$ or $-\infty$. On the graph, you'll see a vertical asymptote. For example, $f(x) = \frac{1}{x}$ has an infinite discontinuity at $x = 0$.

Applications of the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) says: if $f(x)$ is continuous on $[a, b]$, and $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a, b)$ where $f(c) = k$.

In plain terms: a continuous function can't skip over values. If the function goes from 2 to 5, it must hit every number in between at least once.

The IVT is especially useful for proving that equations have solutions (root-finding). Here's how to apply it:

1. Confirm that $f(x)$ is continuous on $[a, b]$
2. Calculate $f(a)$ and $f(b)$
3. Check whether your target value $k$ falls between $f(a)$ and $f(b)$
4. If all conditions hold, conclude that at least one $c$ in $(a, b)$ satisfies $f(c) = k$

Example: Show that $f(x) = x^3 - x - 1$ has a root on $[1, 2]$. Since $f(1) = -1$ and $f(2) = 5$, and $0$ is between $-1$ and $5$, the IVT guarantees some $c$ in $(1, 2)$ where $f(c) = 0$.

Limits of Composite Functions

For a composite function $g(f(x))$, you evaluate the limit from the inside out:

1. Find $\lim_{x \to a} f(x) = L$ (the limit of the inner function)
2. If $g$ is continuous at $L$, then $\lim_{x \to a} g(f(x)) = g(L)$

The key condition is that the outer function $g$ must be continuous at $L$. When $g$ is continuous at the relevant point, you can simply substitute: plug the inner limit into the outer function. This is why continuity matters so much for computation.

More generally, if $\lim_{x \to a} f(x) = L$ and $\lim_{x \to L} g(x) = M$, then $\lim_{x \to a} g(f(x)) = M$, even if $g$ isn't continuous at $L$, as long as $f(x) \neq L$ near $a$.

Graphical vs. Algebraic Continuity Analysis

Graphical analysis is fast for identifying discontinuities visually:

• Look for holes (removable), jumps (jump), or vertical asymptotes (infinite)
• If you can trace the curve through a region without lifting your pencil, the function is continuous there
• Graphs are great for a quick check, but they can be misleading at fine scales

Algebraic analysis gives you a rigorous answer. At each point in question:

1. Compute $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ to see if the two-sided limit exists
2. Check that $f(a)$ is defined
3. Verify that $\lim_{x \to a} f(x) = f(a)$

On exams, algebraic verification is what earns full credit. Use the graph to guide your intuition, then confirm with algebra.

Function Properties and Continuity

Domain and continuity are closely linked. Discontinuities often show up at points where the function is undefined, such as where a denominator equals zero or where a square root has a negative argument. Common continuous functions include polynomials (continuous everywhere), rational functions (continuous on their domain), and trig functions like $\sin(x)$ and $\cos(x)$ (continuous everywhere).

Differentiability implies continuity. If a function is differentiable at a point, it must be continuous there. But the reverse is not true: a function can be continuous at a point without being differentiable. The classic example is $f(x) = |x|$ at $x = 0$. The graph has no break there (continuous), but it has a sharp corner (not differentiable). Think of it as a one-way relationship: differentiability is a stronger condition than continuity.