$(5$ points$)$ True or False

Indicate whether the following are true or false. No justification needed.

For all questions, let $f$ be defined on $(-,).$

a$)$ If $f$ is continuous everywhere then $f$ is integrable on any interval.

b$)$ If $f$ and $g$ are both differentiable at $x,$ and if $g(x)0,$ then $\frac{f}{g}$ is differentiable at $x$ and

$\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{f^{\text{'}}(x)}{g^{\text{'}}(x)}.$

c$)$ If $A(x)={\displaystyle \underset{a}{\overset{x}{}}}f(t)dt,$ then $A^{\text{'}}(x)=\underset{h0}{lim}\frac{{\displaystyle \underset{x}{\overset{x+h}{}}}f(t)dt}{h}.$

d$)$ If $f$ is continuous at $x,$ then $f^{\text{'}}(x)$ exists.

e$)$ If $f^{\text{'}}(x)=g^{\text{'}}(x)$ for every $x,$ then there exists a constant $C$ so that, for every $x,g(x)=f(x)+C.$

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