$20.$ For the piecewise function $f(t)$ given to the right define the antiderivative function $F(x)={\displaystyle \underset{0}{\overset{x}{}}}f(t)dt.$

a$)$ Find the rule for $F(x).$

$f(t)=\{\begin{array}{cc}1,& t<mn>1</mn>\\ 3t^2-1,& 1t2\\ 6t,& t>2\end{array}$

b$)$ Graph $F(x)$ on the interval $0,3$

c$)$ At what values of $t$ is $f(t)$ discontinuous?

b$)$ The values of $x$ is $F(x)$ discontinuous?

d$)$ At what values of $x$ is $F(x)$ not differentiable? Explain.

$21.$ Let $g$:$RR$ be a function and the sequence $g_n(x)=\frac{1+n*x*g(x)}{n}.$ Prove that the sequence of functions $g_n$ converges uniformly to the function $x*g(x).($i$.$e$.g_n(x)x*g(x)$ uniformly$)$