What is derivative of one conditional function.

Question

$$ F(x) = \cases{x& if $x>\lambda$\\ 0& if $x<\lambda$} $$ What is partial derivative of $f(x, lambda)$ with respect to $\lambda$?

Answer 1

$$f(x,\lambda) = \cases{x& if $x>\lambda$\\ 0& if $x<\lambda$}$$

$$ \frac {\partial f}{\partial \lambda } =0 \text { if } x\ne \lambda$$

$$ \frac {\partial f}{\partial \lambda } = \text { does not exist, if } x= \lambda$$

Answer 2

Hint: It should be written like that: $f(x, \lambda)=\frac{x+xsgn(x-\lambda)}{2}$.

Also note that derivative of sign function is twice Dirac delta.