Constant functions are analytic.

Question

A function f is said to be analytic if it can be locally represent by power series expansion or equivalently it converges to its Taylor series expansion for each value of its domain.Now consider constant function f(x)=c,for some constant c in it cannot have expansion as all its derivatives becomes zero so I think it is not analytic but my told me it is analytic,can anybody explain whether constant function is analytic or not? Thanks in advance

Answer 1

Consider the power series $$c+0x+0x^2+0x^3+\dots$$

Alternatively, putting $f(x)=c$, it is easy to see that the complex limit

$$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} = 0$$

hence the constant function $f$ is analytic with derivative $0$.