Continuous Real Function/Examples/Sine of x over x with 1 at 0

Examples of Continuous Real Functions

Let $f: \R_{\ge 0} \to \R$ be the real function defined as:

$\map f x = \begin {cases} \dfrac {\sin x} x & : x \ne 0 \\ 1 & : x = 0 \end {cases}$

Then $\map f x$ is continuous at $x = 0$.

Proof

From Limit of $ \dfrac {\sin x} x$, we have that:

$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$

The result follows by definition of continuous real function.

$\blacksquare$

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