Smooth vs. Non-smooth Functions

Smooth functions have a unique defined first derivative (slope or gradient) at every point. Graphically, a smooth function of a single variable can be plotted as a single

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vector analysis - What is a good metric for the smoothness of a $N$ dimensional polynomial? - Mathematics Stack Exchange

Smooth and non-smooth control-Lyapunov function

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On a new smoothing technique for non-smooth, non-convex optimization

The derivative of a non-smooth function is set valued at