Proving that if $f\in\mathscr{R} [a,b]$ then $\sqrt{f}\in \mathscr{R}[a,b]$

Question

I'm trying to show that assuming $f:[a,b] \rightarrow \mathbb{R}$ is nonnegative, bounded function that is Riemann integrable on [a,b], then $\sqrt{f}$ is Riemann integrable on [a,b].

I know we have to construct an $\epsilon$ proof using the fact that f is Riemann integrable. We know that $\exists P$, a partition, such that $U(f,p) - L(f,p) < \epsilon$.

And we want to show that $| \sqrt{f(x)} - \sqrt{f(y)}|$ is somehow less than $\sum (\sup_{[x_i-1, x_i]} f - \inf_{[x_i-1, x_i]} f) \Delta x_i$, which in turn will be less than $\epsilon$. I can't seem to figure this bit out, so a hint or help would be much appreciated! Here's what I've tried (but I'm not sure if it's correct):

$$| \sqrt{f(x)} - \sqrt{f(y)}| \leq |f(x) - f(y)| \leq \sup_{[x_{i-1}, x_i]} f - \inf_{[x_{i-1}, x_i]} f \text{ } (\forall 1 \leq i \leq N) $$

And so $\exists$ partition P such that:

$$U(\sqrt{f}, p) - L (\sqrt{f}, p) \leq U(f, p) - L (f,p) < \epsilon. $$

Answer 1

Note that $|\sqrt{f(x)}-\sqrt{f(y)}|$ is not necessarily smaller than $|\,f(x)-f(y)|$, if $\,f$ is less that $1$.

But, $$ |\sqrt{x}-\sqrt{y}|=\frac{|x-y|}{|\sqrt{x}+\sqrt{y}|}\le \frac{|x-y|}{\max\{\sqrt{x},\sqrt{y}\}}. $$ Let $\varepsilon>0$. As $f$ is Riemann integrable over $[a,b]$ we may choose a partition $P=\{t_0<\cdots<t_n\}$ of $[a,b]$, with $\,m_i,\,M_i$, the infima and suprema of $f$ in the subinterval $[t_{i-1},t_i]$, such that $U(\,f,P)-L(\,f,P)<\varepsilon^2$, then $$ U(\sqrt{f},P)-L(\sqrt{f},P)=\sum_{i=1}^n(\sqrt{M_i}-\sqrt{m_i})(t_i-t_{i-1})\\=\sum_{M_i\ge\varepsilon^2}(\sqrt{M_i}-\sqrt{m_i})(t_i-t_{i-1})+ \sum_{M_i<\varepsilon^2}(\sqrt{M_i}-\sqrt{m_i})(t_i-t_{i-1}) \\ \le \frac{1}{\varepsilon}\sum_{i=1}^n(M_i-m_i)(t_i-t_{i-1})+\varepsilon\sum_{i=1}^n(t_i-t_{i-1})<\varepsilon+\varepsilon(b-a). $$ Since $$ \sum_{M_i\ge\varepsilon^2}(\sqrt{M_i}-\sqrt{m_i})(t_i-t_{i-1})\le \sum_{M_i\ge\varepsilon^2}\frac{1}{\sqrt{M_i}}(M_i-m_i)(t_i-t_{i-1})\le\frac{1}{{\varepsilon}}\big(U(\,f,P)-L(\,f,P)\big)<\varepsilon, $$ while $$ \sum_{M_i<\varepsilon^2}(\sqrt{M_i}-\sqrt{m_i})(t_i-t_{i-1})\le \sum_{M_i<\varepsilon^2}\varepsilon(t_i-t_{i-1})\le \varepsilon(b-a). $$

Answer 2

In fact, if $f$ is Riemann integrable on $[a,b],$ $f([a,b]) \subset [c,d],$ and $g$ is continuous on $[c,d],$ then $g\circ f$ is Riemann integrable on $[a,b].$ Your problem is a special case of this.