I've been asked to show where the following function is analytic and differentiable;
$$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$
First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$.
Then, I verified the Cauchy-Riemann equations; $$u_y = -v_x \implies 0 = 0$$ $$u_x = v_y \implies 4x^3 = 4(y-1)^3 \implies x = y-1$$
Would it then be correct to say that $f$ is analytic where $x = y-1$, but differentiable on all of $\mathbb{C}$??
I'm just a bit confused about what we've been taught in class as to the explicit definitions of an analytic function and a differentiable function, so I guess I'm just asking for some clarification on these terms, using this question as an example.
Any input would be fantastic!!