Exercise 7.5.2

Question

Decide whether each statement is true or false, providing a short justification for each conclusion.
\begin{enumerate}[label=(\alph*)] 
\item If $g=h^{\prime}$ for some $h$ on $[a, b]$, then $g$ is continuous on $[a, b]$.
\item If $g$ is continuous on $[a, b]$, then $g=h^{\prime}$ for some $h$ on $[a, b]$.
\item If $H(x)=\int_{a}^{x} h$ is differentiable at $c \in[a, b]$, then $h$ is continuous at $c$.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item False, e.g. $h(x) = x^2 \sin (1/x)$ as explored in section 5.1.
\item True; if $g$ is continuous then it is also integrable, and by the Fundamental Theorem of Calculus we have that $G(x) = \int^x_a g$ is differentiable over $[a,b]$ with $G' = g$.
\item False; consider Thomae's function $t(x)$. From Exercise 7.3.2 we have that $t$ is integrable with $\int^1_0 t = 0$; it's also easy to show that $\int^x_0 t = 0$. Then $H(x) = 0$ is differentiable over $[0,1]$, but $t(x)$ is not continuous over $[0,1]$.
 \end{enumerate}

Exercise 7.5.2

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Exercise 7.5.2

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