Continuity & Types of Discontinuity Cheat Sheet

Key Facts

• A function $f$ is continuous at $x=a$ if $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x)=f(a)$.
• The two-sided limit $\lim_{x \to a} f(x)$ exists only when $\lim_{x \to a^-} f(x)=\lim_{x \to a^+} f(x)$.
• A removable discontinuity occurs when $\lim_{x \to a} f(x)$ exists but $f(a)$ is undefined or $f(a) \ne \lim_{x \to a} f(x)$.
• A jump discontinuity occurs when $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ both exist but are not equal.
• An infinite discontinuity occurs when at least one one-sided limit is infinite, such as $\lim_{x \to a^+} f(x)=\infty$ or $\lim_{x \to a^-} f(x)=-\infty$.
• A rational function may have a removable discontinuity where a common factor cancels, such as $f(x)=\frac{x^2-4}{x-2}=x+2$ for $x \ne 2$.
• Polynomial functions are continuous for all real numbers, and rational functions are continuous wherever their denominators are not zero.
• Piecewise functions are checked for continuity at boundary points by comparing $\lim_{x \to a^-} f(x)$, $\lim_{x \to a^+} f(x)$, and $f(a)$.

Vocabulary

Continuity

A function is continuous at $x=a$ when the graph has no break there and $\lim_{x \to a} f(x)=f(a)$.

One-sided limit

A one-sided limit describes the value a function approaches from only one side, written $\lim_{x \to a^-} f(x)$ or $\lim_{x \to a^+} f(x)$.

Removable discontinuity

A removable discontinuity is a hole or misplaced point where $\lim_{x \to a} f(x)$ exists but does not match the function value.

Jump discontinuity

A jump discontinuity occurs when the left-hand and right-hand limits at $x=a$ exist but have different values.

Infinite discontinuity

An infinite discontinuity occurs when a function grows without bound near $x=a$, often because of a vertical asymptote.

Oscillating discontinuity

An oscillating discontinuity occurs when function values keep fluctuating near $x=a$ so no single limiting value exists.