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The equation $$f(x + \epsilon) \approx f(x) + \epsilon f'(x)$$ is justified from taylor series. It is not derived from the limit definition of a derivative. Let $f'(a)$ and $f''(a)$ exist for $a$ in the interval $(x, x+\epsilon)$. Then, \begin{align*} f(x + \epsilon) = f(x) + \epsilon f'(x) + \frac{\epsilon^2}{2}f''(b) \end{align*} where $b$ is some number ...

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From reading the text, it's clear that the authors are using critical point to mean the same thing as stationary point, so they are not using the proper mathematical definition. More generally, automatic differentiation will return a gradient at nondifferentiable points. Ask tensorflow or pytorch to take the gradient of $\text{ReLU}(x)\big|_{x=0}$. They ...

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A critical point of a function $f$ can be a stationary point (i.e. $f'(x) = 0$), or a point where the derivative is undefined (for example, in the case of the absolute value function $f(x)$, $x=0$ is a critical point, as $f$ is not differentiable at $x=0$). So, all stationary points are critical points. These notes provide more examples of how to find ...

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It is just an assumption. If we assume $\epsilon$ is small enough (depending on the function $f$), you can remove the limit $\lim_{\epsilon \to 0}$ for the approximation.

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