Continuity

Formal definitions¶

A function $f$ is continuous at $x = a$ if $\\lim_{x \to a} f(x) = f(a)$.

A function $f$ is right-continuous at $x = a$ if $\\lim_{x \to a^+} f(x) = f(a)$.

A function $f$ is left-continuous at $x = a$ if $\\lim_{x \to a^-} f(x) = f(a)$.

A function $f$ has a jump discontinuity at $x = a$ if the left-hand limit $\\lim_{x \to a^-} f(x)$ and the right-hand limit $\\lim_{x \to a^+} f(x)$ both exist but are not equal.

A function $f$ has a removable discontinuity at $x = a$ if the limit $\\lim_{x \to a} f(x)$ exists but does not equal $f(a)$.