# Continuous Real Function/Examples/Sine of Reciprocal of x

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## Examples of Continuous Real Functions

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

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

Then $\map f x$ is **not** continuous at $x = 0$.

## Proof

From Limit of Real Function: $\map \sin {\dfrac 1 x}$ at $0$, $\map f x$ has no limit at $x = 0$.

In order to be continuous at $x = 0$, it needs to have a limit at $x = 0$.

The fact that $\map f x$ is defined at $x = 0$ is immaterial.

Hence the result.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity: Example $1.4.2$