Exercise 7.5.11

Question

Assume $f$ is integrable on $[a, b]$ and has a "jump discontinuity" at $c \in(a, b)$. This means that both one-sided limits exist as $x$ approaches $c$ from the left and from the right, but that
$$
\lim _{x ightarrow c^{-}} f(x) 
eq \lim _{x ightarrow c^{+}} f(x) .
$$
(This phenomenon is discussed in more detail in Section 4.6.)
\begin{enumerate}[label=(\alph*)] 
\item Show that, in this case, $F(x)=\int_{a}^{x} f$ is not differentiable at $x=c$.
\item The discussion in Section $5.5$ mentions the existence of a continuous monotone function that fails to be differentiable on a dense subset of $\mathbf{R}$. Combine the results of part (a) with Exercise 6.4.10 to show how to construct such a function.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] \item For conciseness, define $l = \lim_{x \to c^-} f(x)$ and $r = \lim_{x \to c^+} f(x)$ as the left and right limits of $f(c)$. Define $$g(x) = \begin{cases} f(x) & x < c \ l & x = c \ f(x) - r + l & x > c \end{cases}$$ where we've essentially shifted $f$ after $x > c$ to line up with $x < c$, so that $g$ is continuous at $c$. As a result, $G(x) = \int^x_a g$ is differentiable at $c$. Now if $F = \int^x_a f$ was also differentiable at $c$, then we would also have $G - F = \int^x_a g - f$ differentiable at $c$. But actually evaluating $\int^x_a g-f$ leaves us with $$ \int^x_a g - f = \begin{cases} 0 & x \leq c \ (l-r)(x-c) & x > c \end{cases} $$ which is not differentiable at $c$. \item Define $h$ and $u_n$ similarly as is done in Exercise 6.4.10, except we only enumerate over the rational numbers in $[0,1]$. Given that $h$ converges uniformly, and each $u_n$ is integrable over $[0,1]$, we have $h$ integrable over $[0,1]$ as well. From part (a) we have that $g(x) = \int^x_a h$ is not differentiable at all rational points (dense in $[0,1]$), while Theorem 7.5.1 (ii) indicates $g$ is continuous. Since $h$ is nonnegative, $g$ must be monotone increasing. We can then extend $g$ over all of $\mathbf{R}$ by repeating it and applying appropriate offsets. \end{enumerate}

Exercise 7.5.11

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Exercise 7.5.11

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