Continuity of Functions & Uniform Convergence

Question

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 $x \in I$.

1. $(a)$ 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?

2. 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?

3. 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 $D_\rho$.

4. If $0\lt r\lt\rho$, then this series converges uniformly in $D_r$.

Answer 1

Hints:

$1.$ $(a)$ Consider $g_n(x)=x^n$ on the interval $[0,1]$. $(b)$ If the convergence is uniform, then $g(x)$ will be continuous. To prove this, remember what is the definition of continuous, and what is the definition of uniform convergence.

$2.$ When you choose your epsilon's and delta's in the definition of uniform convergence, notice that the quantity $\int_a^b \epsilon dx=\epsilon(b-a)$ can be made arbitrarily small.

$3.$ Think of a very simple example.

$4.$ Everything converges absolutely on the boundary $|z|=r$ so we can compare the sum evaluated at points inside to the ones on the boundary to prove it is Cauchy.