Before I ask the question, I know for sure this following question makes sense:
Prove that a necessary condition for $ f(z) $ to be analytic is that the Cauchy-Riemann equations are satisfied.
This question makes sense because if the Cauchy-Riemann equations are satisfied, $f(z)$ need not be analytic. Hence, why it's called a necessary condition.
But in order for $ f(z) $ to be analytic , there are 2 sufficient conditions, the Cauchy-Riemann equations must be satisfied and the partial derivatives must be continuous.
Now for the main question I want to ask:
Prove that a sufficient condition for $ f(z) $ to be analytic is that the Cauchy-Riemann equations are satisfied.
I'm asking this because I had it on my exam yesterday, now how could this make sense, the Cauchy-Riemann equations being satisfied should not be enough, the question did not mention the continuity of the partial derivatives, thus I treated this question as if it was asking "Prove that a sufficient condition for the Cauchy-Riemann equations to be satisfied is that $f(z)$ must be analytic".
So am I right and should I argue with my professor about that?
My question is not about when a function is holomorphic, my question is if the following makes sense as a proof question or not:
Prove that a sufficient condition for f(z) to be analytic is that the Cauchy-Riemann equations are satisfied.
There is a diffrence between differentiability at a point and differentiability/analyticity in a neighborhood of a point. For example $f(x+iy)=\sqrt {|xy|}$ satisfies C-R equations at $0$ but $f$ is not differentiable at $0$. However, in an open set analyticity is equivalent to validity of C-R equations. Continuity of partial derivatives need not be assumed. In fact C-R equations imply that $f$ has continuous partial derivatives of all orders.