If a power series converges uniformly to f on some open U and f in C1 on R, does the series converge everywhere?

Question

Now let me be more precise with my question:

If $\sum c_kx^k = f(x)$ for $x\in U$, where $U$ is some open disc, and if furthermore $\sum c_kx^k$ converges uniformly on $U$ and $f(x)\in C^1(R)$, then does it follow the the series converges uniformly on $R$?

I assumed that it did, since $f$ has a unique power series representation so that $\{c_k\}$ must be the coefficients of its power series. I feel like I might be making a mistake though.

Answer 1

You can define a $C^1$ function $f$ on $\mathbb{R}$ that is equal to $\sum_{n=0}^\infty x^n$ on $(-1/2,1/2)$ (where it converges uniformly), but $\sum x^n$ doesn't converge on $\mathbb{R}$.