Question

examine the differentiability of f(z)=z+1
a)Differentiable everywhere
b)Differentiable at z=1
c)Differentiable at z=0 and z=1
d)Nowhere differentiable​

Answer 1

Answer:

piecewise function is differentiable at a point if both of the pieces have derivatives at that point, and the derivatives are equal at that point. In this case, Sal took the derivatives of each piece: first he took the derivative of x^2 at x=3 and saw that the derivative there is 6.

you take h to be imaginary, f(z+h)=f(z) and the quotient is 0. The limit as h→0 doesn't exist: it can't be both 1 and 0. Thus we say f′(z) doesn't exist, and the function is not differentiable. Another way to see it, it is that the real part of a complex number can be written with its conjugate: Re(x)=12(x+x∗).