$f$ is differentiable on $U\setminus\{p_1,\dots,p_r\}\implies$ $f$ is holomorphic on $U$ 
Let $U\subset\mathbb C$ be open and $p_1,\dots,p_r$ be finite number of points in $U$. If $f:U\to\mathbb C$ is continuous function that is complex-differentiable in any point of $U\setminus\{p_1,\dots,p_r\}$, prove that $f$ is holomorphic on $U$

Since holomorhic is equivalent to being analytic, I wanted to ask if the following is sufficient for a proof.
-If $f$ is continuous on the starklike region $\Omega$ and analytic on $\Omega\setminus\{z_0\}$, then $f$ has a primitive on $\Omega$
-If $f$ has a primitive on starlike region $\Omega$ then $f$ is analytic on $\Omega$
Is $U\setminus\{p_1,\dots,p_r\}$ a starlike region and $f$ is supposed to be continuous only on $U\setminus\{p_1,\dots,p_r\}$, if I could show that $f$ is continuous on $U$, the proof would work ?
If not, can you suggest another method ?
 A: Since your set $\{p_1,\dots,p_r\}$ is finite, let $D_i$ be a disc around each point $p_i$ in your set that doesn't contain the other points (you'll always be able to find a neighborhood around each point that doesn't contain the other points). So  $f$ is holomorphic on a punctured neighborhood $D_i\setminus \{p_i\}$ around each point and continuous on the disc $D_i$.
Now since $f$ is holomorphic on the punctured neighborhood $D_i\setminus \{p_i\}$ , if you consider $f_r$, which is $f$ restricted to $D_i\setminus \{p_i\}$, $f_r$ has a removable singularity at $p_i$. So $f_r$ has a holomorphic extension $F_r$ to $D_i$. Now $F_r$ has a Taylor series centered at $p_i$ that converges to $F_r$ everywhere on $D_i$.
In particular the Taylor series converges to $f_r = f$ everywhere on $D_i\setminus \{p_i\}$. By uniqueness of Laurent series in an annulus, the Laurent series of $f$ on $D_i\setminus \{p_i\}$ is given by a Taylor series:
$$\sum_{n=0}^{\infty} c_n(x-p_i)^n$$
Now use continuity at $p_i$ to show that the Taylor series of $f$ centered at $p_i$ converges to $f$ everywhere on  $D_i$. This means that $f$ is holomorphic on discs around your $p_i$s.
