Example:A smooth function that is not real analytic

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Example

Let [ilmath]f:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] be defined as follows:

• [math]f:x\mapsto\left\{\begin{array}{lr}e^{-\frac{1}{x} } & \text{if }x>0\\ 0 & \text{otherwise}\end{array}\right.[/math]

We will show this function is not real-analytic at [ilmath]0\in\mathbb{R} [/ilmath] but is smooth.