# Sine of X over X is not Continuous at 0

## Theorem

Let $f$ be the real function defined as:

$\map f x := \dfrac {\sin x} x$

Then $f$ is not continuous at $x = 0$.

## Proof

For $f$ to be continuous at $x = 0$ it is necessary that it be defined there.

But at the point $x = 0$, we have that $\map f x = \dfrac {\sin 0} 0$.

Division by $0$ is not defined.

Hence $f$ is not continuous at $x = 0$.

$\blacksquare$