12.3 Continuity

Continuity

Continuity describes whether a function has smooth, unbroken behavior at a point or across an interval. A continuous function is one you could draw without lifting your pencil. Understanding continuity matters because it's a foundational requirement for the calculus concepts you're building toward, especially derivatives and integrals.

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Points of continuity and discontinuity

A function is continuous at a point $x = a$ only if it satisfies all three of these conditions:

1. $f(a)$ is defined — the function actually has a value at $x = a$ (no holes or gaps in the domain).
2. $\lim_{x \to a} f(x)$ exists — the limit from the left and the limit from the right both converge to the same single value.
3. $\lim_{x \to a} f(x) = f(a)$ — the limit equals the actual function value (no jumps or mismatches).

If any one of these fails, the function is discontinuous at that point.

When you're scanning a function for potential discontinuities, focus on these trouble spots:

• Domain restrictions — places where you'd divide by zero, or take an even root of a negative number
• Piecewise definitions — wherever the formula changes, check that the pieces connect smoothly
• Endpoints of intervals — here you can only check one-sided continuity (from the left or from the right, not both)

Evaluating continuity at specific values

To determine whether a function is continuous at a specific point $x = a$, follow these steps:

1. Check that $f(a)$ is defined. Plug $a$ into the function. If $a$ isn't in the domain (e.g., it causes division by zero), stop here — the function is discontinuous.

2. Evaluate the left-hand and right-hand limits.

   • Find $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$.
   • If both one-sided limits exist and are equal, then $\lim_{x \to a} f(x)$ exists. If they differ, the overall limit does not exist, and the function is discontinuous.

3. Compare the limit to the function value. If $\lim_{x \to a} f(x) = f(a)$, the function is continuous at $x = a$. If they don't match, it's discontinuous.

Types of function discontinuities

There are four main types of discontinuity you should be able to identify.

Removable discontinuity (hole) — The limit exists at $x = a$, but either $f(a)$ is undefined or $f(a)$ doesn't equal the limit. You can "fix" this by redefining $f(a)$ to match the limit.

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$. Factor the numerator to get $\frac{(x-1)(x+1)}{x-1}$, which simplifies to $x + 1$ for all $x \neq 1$. So $\lim_{x \to 1} f(x) = 2$, but $f(1)$ is undefined. There's a hole at $(1, 2)$.

Jump discontinuity — The left-hand and right-hand limits both exist but aren't equal. The function "jumps" from one value to another.

• Example: $f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases}$ has a jump at $x = 0$ because $\lim_{x \to 0^-} f(x) = 1$ while $\lim_{x \to 0^+} f(x) = 2$.

Infinite discontinuity (vertical asymptote) — The function blows up toward $+\infty$ or $-\infty$ as $x$ approaches $a$ from one or both sides.

• Example: $f(x) = \frac{1}{x}$ at $x = 0$. From the right the function heads to $+\infty$, and from the left it heads to $-\infty$. The limit doesn't exist as a finite number.

Oscillating discontinuity — The function bounces between values near $x = a$ so rapidly that no single limit is approached.

• Example: $f(x) = \sin\!\left(\frac{1}{x}\right)$ at $x = 0$. As $x$ gets closer to 0, the function oscillates between $-1$ and $1$ faster and faster, so neither one-sided limit exists.

Continuity and Differentiability

These two concepts are closely linked, but the relationship only goes one direction.

• Differentiability implies continuity. If a function is differentiable at a point, it must be continuous there. A function can't have a well-defined slope at a point where it has a gap or jump.
• Continuity does NOT imply differentiability. A function can be continuous at a point but still fail to be differentiable there. The classic example is $f(x) = |x|$ at $x = 0$: the function is continuous (no breaks), but it has a sharp corner, so no single tangent line exists.

This is a distinction that shows up frequently on tests: every differentiable function is continuous, but not every continuous function is differentiable.