Example:A smooth function that is not real analytic

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Example

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined as follows:

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

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