Continuous Real Function/Examples/Sine of Reciprocal of x

Examples of Continuous Real Functions

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

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

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 $1.4.2$