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 $x = 0$.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity