1. The problem statement, all variables and given/known data Determine where the function f(x + iy) = 2sin(x) + iy^2 + 4(ix - y) is differentiable and where it is analytic. 3. The attempt at a solution f(x + iy) = 2sin(x) -4y + i(y^2 +4x) Through C-R equations: du/dx = 2 cos x dv/dy = 2y du/dy = -4 dv/dx = 4 So the C-R equations hold only if y = cos (x) 1) Supposedly these partial derivatives are continuous everywhere, I do not understand what this exactly means, what is an example of a partial derivative that is not continuous everywhere? Also it says f is differentiable only on the curve y = cos (x), but it's not analytic since it's not differentiable in the e-neighbourhood at a point on the curve. 2) I do know that for a function to be analytic it must be differentiable at a given point, and also in the e-neighbourhood of it, but how is this exactly determined, and what does it mean to be differentiable in an e-neighbourhood of a point in this curve specifically? If I draw out y = cos (x), pick out a point and look at the e-neighbourhood around this point, all you get is the point on that line, a portion of the line going through, and nothing else around it.