What does mean ‘regular function at the origin’?

Question

Really I'm new here , and I seek for simple definition about regular function at

the origin when the function is presented as a power series ?

Answer 1

The meaning of the word regular is not precisely defined. Sometimes they say regular enough which means (for instance) that a function is differentiable, or twice continuously differentiable and so on. Usually saying this they want the function to fulfill all the needed assumptions.

In your context, if a function is representable as a power series $$f(x)=\sum_{n=0}^{\infty}a_nx^n$$ centered at zero, this series converges in a certain interval. In this interval $f$ is infinitely times differentiable. It is also possible that the power series is convergent only at $x=0$. In this case nothing could be said about $f$ at the other points. Take into account the series $$\sum_{n=0}^{\infty}n^nx^n.$$ It is divergent for any $x\ne 0$ and the function defined by this series is $f\colon\{0\}\to\{a_0\}$, $f(0)=a_0.$ Such function is, of course, not differentiable. It is not possible to compute any limit connected with $f$.