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What is the derivative of the ReLU activation function defined as:

$$ \mathrm{ReLU}(x) = \mathrm{max}(0, x)$$

What about the special case where there is a discontinuity in the function at $x=0$?

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The derivative is:

$$ f(x)= \begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{if } x > 0 \\ \end{cases} $$

And undefined in $x=0$.

The reason for it being undefined at $x=0$ is that its left- and right derivative are not equal.

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    $\begingroup$ So in practice (implementation), one just picks either $0$ or $1$ for the $x=0$ case? $\endgroup$ – Tom Hale Mar 14 '18 at 9:51
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    $\begingroup$ The convention is that drdx=1(x>0) $\endgroup$ – neuroguy123 Mar 14 '18 at 13:10
  • $\begingroup$ @TomHale why not use $f(0) = \frac{1}{2}$? The practical implementation is another matter and worth a new question. This is just the mathematics. $\endgroup$ – Jim Mar 14 '18 at 16:17
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    $\begingroup$ @TomHale by the way, see Nouroz Rahman's answer at quora.com/…: "[...] In my view, in built-in library functions (for example: tf.nn.relu()) derivative at x = 0 is taken zero to ensure a sparser matrix..." $\endgroup$ – Jim Mar 29 '18 at 16:17

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