How to show a function is locally C^1 implies globally C1?

Question

Actually there is a series problem like f(x)=sum(n=1 to ∞)[sin(nx^2)/1+n^3], the question was whether f(x) is C^1 or not. This question has already answered, but a big issue of mine is I can't find any hint from those. There were told It is enough to show f(x) is C^1 locally and hence it follows globally. But how? I'm also aware of the the theorem Rudin page 152 Theorem 7.17: Suppose  a sequence of functions, differentiable on [a, b] and such that {fn}  converges for some point  on [a,b]. If  {f'n}  converges uniformly on [a,b], then {fn} converges uniformly on [a,b] to f and lim(f'n)=f',for all a<= x<=b But I don't know how it follows from this theorem. Please help me find it out. Thanks in advance.

Answer 1

$f\in C^{1}$ means $f$ is differentiable and its derivative is continuous. This is same as saying that $f$ is differentiable at each point and its derivative is continuous at each point. To prove this fix a point $x$. If you know that $f$ is a $C^{1}$ function in some interval around $x$ then $f'$ is continuous at $x$ so we are done.