The mistake is that you are confusing differenciable as used in real analysis and differenciable as used in complex analysis. It is the same word but it means different things!I have found a question
Prove that f(z)=Re(z) is not differentiable at any point.
According to me f(z)=Re(z)=Re(x+iy)=x which is differentiable everywhere. Then where is the mistake?