Let

$f(x)=\{\begin{array}{c}x\text{;}(0x<mn\; id="EditorElement\_747699">1</mn><mi\; id="EditorElement\_747701">)</mi>\\ 2x\text{;}(1x2)\end{array}$

i$)$ Is the function $f(x)$ continuous on $0,2?$ Explain $($a graph is not a proof!$)$

ii$)$ Compute AND graph the function

$F(x)={\displaystyle \underset{0}{\overset{x}{}}}f(t)dt$

Iam done with the first two parts, I will need help with the next parts

iii$)$ Is the function F$($x$)$ continuous at x$=1$

iv$)$ Does F $(1)$ exist? Explain. $($Hint: Compute the derivative of F $($x$)$ from the left and right at x $=1)$

iv$)$ Does iii$)$ violate the FTC II that states F $($x$)=$ f $($x$)$ for all x at which f is continuous?

In the second part, when we are calculating F$($x$),$ for the interval $1$ to $2,$ we should be considering two intervals $0$ to $1$ and $1$ to $2.$