Continuous Real Function/Examples

Examples of Continuous Real Functions

Example: $\sqrt x$ at $x = 1$

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

$\map f x = \sqrt x$

Then $\map f x$ is continuous at $x = 1$.

Example: $\dfrac {\sin x} x$ with $1$ at $x = 0$

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 $x = 0$.

Example: $\map \sin {\dfrac 1 x}$ with $0$ at $x = 0$

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 $x = 0$.