I am trying to solve the following problems. Do you have any hints?
Let $g_n(x)$ be functions defined on interval $I = [a,b]$ and suppose $g(x) =\lim_{n \to \infty} g_n(x)$ is defined for every $x \in I$.
$(a)$ If every $g_n(x)$ is continuous, does it follow that $g(x)$ is continuous?
$(b)$ If that convergence is uniform in $I$, is the limit function $g$ is continuous?If the convergence is uniform, does it follow that the limit of the integral of $g_n(x)$ from $a$ to $b$ is the integral of the limit?
If $\sum_{n \geq 0} c_n X^n$ has radius of convergence $\rho >0$, then $g(z) = \sum_{\geq 0}c_n z^n$ is defined in the disk $D_\rho = \{z: |z|\lt p\}$ / Give an example that does not converge uniformly in $D_\rho$.
If $0\lt r\lt\rho$, then this series converges uniformly in $D_r$.