# Does this question about the sufficiency of the analyticity of f(z) make sense [duplicate]

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.

## marked as duplicate by SmileyCraft, KReiser, Lord Shark the Unknown, Leucippus, José Carlos Santos complex-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jan 7 at 8:40

• I may have misunderstood your question then. Does this answer your question? "The function given by $f(z)=e^{-z^{-4}}$ for $z\neq0$, $f(0)=0$ satisfies the Cauchy–Riemann equations everywhere but is not analytic (or even continuous) at $z=0$." en.wikipedia.org/wiki/Looman%E2%80%93Menchoff_theorem – SmileyCraft Jan 6 at 23:31
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.
• So that means that the question should still mention that the function is differentiable everywhere, correct? So that the cauchy riemann equations when satisfied, imply the analyticity of $f(z)$ 100 % (even at zero) – khaled014z Jan 6 at 23:49