Question

If a function is analytic in a domain, then it is necessarily:

• A. Continuous only
• B. Differentiable only once
• C. Infinitely differentiable
• D. Bounded

Hint:

In complex analysis, remember: {Analytic $\Rightarrow$ infinitely differentiable $\Rightarrow$ power series expansion}.

Answer 1

The Correct Option is C

Step 1: Understanding the meaning of an analytic function.
A complex function is said to be analytic in a domain if it is complex differentiable at every point of that domain. Complex differentiability is a much stronger condition than real differentiability.
Step 2: Important property of analytic functions.
One of the fundamental results of complex analysis states that if a function is analytic in a domain, then it possesses derivatives of all orders in that domain. That is, the function is infinitely differentiable.
Step 3: Theoretical justification.
Analytic functions satisfy the Cauchy–Riemann equations and can be represented by a power series in a neighborhood of every point in the domain. This power series representation guarantees the existence of derivatives of all orders.
Step 4: Analysis of the given options.
(A) Continuous only: Incorrect — analytic functions are not just continuous, they have much stronger smoothness properties.
(B) Differentiable only once: Incorrect — analyticity implies differentiability of all orders.
(C) Infinitely differentiable: Correct — this is a direct consequence of analyticity.
(D) Bounded: Incorrect — an analytic function need not be bounded in its domain.
Step 5: Conclusion.
Since every analytic function has derivatives of all orders in its domain, it is necessarily infinitely differentiable.