## Types of Discontinuity ![[Pasted image 20220516184209.png]] If $f(x)$ is discontinuous at point $a$, then: 1. $f$ has a **removable discontinuity** *(hole)* at $a$ if $\lim_\limits{x \to a} f(x)$ exists. - *Note:* When we state that $\lim_\limits{x \to a} f(x)$ exists, we mean that $\lim_\limits{x \to a} f(x) = L$, where $L$ is a real number. 2. $f$ has a **jump discontinuity** at $a$ if $\lim_\limits{x \to a^-} f(x)$ and $\lim_\limits{x \to a^+} f(x)$ exist, but $\lim_\limits{x \to a^-} f(x) \ne \lim_\limits{x \to a^+}$ - **Note:** *When we state that $\lim_\limits{x \to a^-} f(x)$ and $\lim_\limits{x \to a^+}$ both exist, we mean that both limits are real-valued and that neither take on the values $\pm \infty$.* 3. $f$ has an **infinite discontinuity** (*[[Asymptotes]]*) at $a$ if $\lim_\limits{x \to a^-} f(x) = \pm \infty$ and/or $\lim_\limits{x \to a^+} f(x) = \pm \infty$ - **Note:** *Implies a vertical asymptote.*