# How can you tell if a function has a discontinuity?

Jun 19, 2015

Let be $f \left(x\right) : A \to B$. The function is discontinuous in $\mathbb{R}$ if and only if $A \subset \mathbb{R}$ and $A \ne \mathbb{R}$.
Pragmatically we can say that if a function has a fractional part, a logarithm, an even root, a tan(x), a cot(x), an arcsin(x), an arccos(x) there might be one or more points of discontinuity. The only way is to verify if there are points where the function isn't defined.
I make some examples:

$a \left(x\right) = \left\{\begin{matrix}x \mathmr{if} x > 2 \\ 2 \text{ if } 2 \ge x > 1\end{matrix}\right.$
$b \left(x\right) = {\log}_{5} \left(3 x\right)$
$c \left(x\right) = \frac{x}{x}$
$d \left(x\right) = \arccos \left(x\right)$
$e \left(x\right) = {e}^{x} / 3$
$f \left(x\right) = \sqrt{\left\mid x \right\mid}$

${D}_{\text{a(x)}} = \left\{x | x \in \mathbb{R} \wedge x > 1\right\}$
${D}_{\text{b(x)}} = \left\{x | x \in \mathbb{R} \wedge x > 0\right\}$
${D}_{\text{c(x)}} = \left\{x | x \in \mathbb{R} \wedge x \ne 0\right\}$
${D}_{\text{d(x)}} = \left\{x | x \in \mathbb{R} \wedge 0 \le x \le \pi\right\}$
${D}_{\text{e(x)}} = \mathbb{R}$
${D}_{\text{f(x)}} = \mathbb{R}$
So e(x) and f(x) are continuous.