Continuous Real Function/Examples
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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 $x = 0$.