Originally Posted by

**heathrowjohnny** Lets say we have a function: $\displaystyle f(z) = x+i|y| $.

We want to find all point that is it differentiable. Right of the bat we know that it is differentiable for $\displaystyle y >0 $. For $\displaystyle f(z) = x+iy $ is a "direct" function of $\displaystyle x+iy $. E.g. we do not have to appeal to CR equations.

Also it is not differentiable for $\displaystyle y<0 $ since $\displaystyle f(z) = x-iy $ is not a "direct" function of $\displaystyle x+iy $.

Now for $\displaystyle y = 0 $ can we use the same argument to show that $\displaystyle f(z) = x+i|y| $ is not differentiable there?

Because then we have $\displaystyle f(z) = x $. And this is not a "direct" function of $\displaystyle x+iy $.