Lets say we have a function of a complex variable z , f(z). I read that for the function to be differentiable at a point z_{0} , the CR equations are a necessary condition but not a sufficient condition. Can someone give me an example where the CR equations hold but the function is not differentiable at that point , thus justifying that the CR equations holding true aren't sufficient test. I am unable to visualise.
Let g(x, y)= 1 if xy is not 0, 0 if xy= 0 and let f(z)= g(x,y)(1+ i)= g(x, y)+ ig(x,y) where z= x+ iy. Then [tex]\frac{g(x,y)}{\partial x}= \frac{\partial g(x,y)}{\partial y}= 0[/tex] for (x,y)= (0, 0) so the Riemann-Cauchy equations are satisfied there but the function is not even continuous at (0, 0).