According to the Royden 4th p.119,
Definition) A real-valued function $f$ on a closed, bounded interval $[a,b]$ is said to be absolutely continuous on $[a,b]$ provided for each $\epsilon >0$, there is a $\delta>0$ such that for every finite disjoint collection $\{ (a_k,b_k)\}_{k=1}^{n} $ of open intervals in $(a,b)$,
if $\sum_{k=1}^{n}[b_k-a_k] < \delta$, then $\sum_{k=1}^n |f(b_k)-f(a_k) | < \epsilon$.
I am trying to prove that $f(x) = \sqrt{x}$ for $0 \leq x \leq 1$ is absolutely continuous on $[0,1]$.
I want to check that my approach is correct.
Proof)
Claim 1 For given $0<\epsilon<1$, $f(x) = \sqrt{x}$ is absolutely continuous on $[\epsilon,1]$.
Let $0 < \epsilon <1$ and consider finite disjoint collection $\{ (a_k,b_k)\}_{k=1}^{n} $ of open intervals in$(\epsilon,1)$.
Note that $\sum_{k=1}^{n} |f(b_k)-f(a_k)| $ = $\sum_{k=1}^{n}|\sqrt{b_k}-\sqrt{a_k}|$= $\sum_{k=1}^{n} \frac{b_k-a_k}{\sqrt{b_k}+\sqrt{a_k}}$ < $\frac{1}{2\sqrt{\epsilon}}$ $\sum_{k=1}^{n}[b_k-a_k]$.
Let $\delta = 2\epsilon \sqrt{\epsilon} = 2\epsilon^{\frac{3}{2}}$.
Now, if $\sum_{k=1}^{n}[b_k – a_k] < \delta = 2\epsilon^{\frac{3}{2}}$, then $\sum_{k=1}^{n} |f(b_k)-f(a_k)| $ = $\sum_{k=1}^{n} \frac{b_k-a_k}{\sqrt{b_k}+\sqrt{a_k}}$ < $\frac{1}{2\sqrt{\epsilon}} (2\epsilon^{\frac{3}{2}})$ = $\epsilon$.
Thus, $f(x) = \sqrt{x}$ is absolutely continuous on $[\epsilon,1]$. #
Claim 2 $f(x) = \sqrt{x}$ is absolutely continuous on $[0,1]$.
Let $\epsilon>0$ be given and choose $\epsilon'>0$ such that $0<\sqrt{\epsilon'}<\frac{\epsilon}{2}$.
By Claim 1, we can find $\delta>0$ as the response to the $\epsilon'$ challenge regarding the criterion for the absolute continuity of $f(x)=\sqrt{x}$ on $[\epsilon',1]$.
Consider $\{ (a_k,b_k) \}_{k=1}^{n}$ such that $\sum[b_k-a_k]< \delta$.
Divide a disjoint collection $\{ (a_k,b_k) \}_{k=1}^n$ in $(0,1)$ into two parts:
$\{ (a_{k1},b_{k1}) \}_{k1=1}^{n1}$ in $(0,\epsilon')$ and $\{ (a_{k2},b_{k2}) \}_{k2=1}^{n2}$ in $(\epsilon',1)$, where $n1+n2=n$.
(If some $(a_k,b_k)$ contains $\epsilon'$, then divide it into two parts: $(a_k,\epsilon')$ and $(\epsilon',b_k$).)
Now, $\sum_{k2=1}^{n2}[b_{k2}-a_{k2}] < \sum_{k=1}^{n}[b_{k}-a_{k}] < \delta$.
Since $\delta>0$ responds to the $\epsilon'>0$ challenge on $[\epsilon',1]$, $\sum_{k2=1}^{n2} |\sqrt{b_{k2}} – \sqrt{a_{k2}}| < \epsilon' < \frac{\epsilon}{2}$.
Also, $\sum_{k1=1}^{n1} |\sqrt{b_{k1}} – \sqrt{a_{k1}}| < \sqrt{\epsilon'}-0 < \frac{\epsilon}{2}$.
Thus, $\sum_{k=1}^{n} |\sqrt{b_{k}} – \sqrt{a_{k}}| \leq$ $\sum_{k1=1}^{n1} |\sqrt{b_{k1}} – \sqrt{a_{k1}}|$ + $\sum_{k2=1}^{n2} |\sqrt{b_{k2}} – \sqrt{a_{k2}}|$ < $\frac{\epsilon}{2}$ + $\frac{\epsilon}{2}$ < $\epsilon$.
Hence, $f(x) = \sqrt{x}$ is absolutely continuous on $[0,1]$. #
Best Answer
Here is a proof that uses properties of the integral.
Note that $\sqrt x= \int_0^x \frac{1}{2\sqrt{t}} dt$ for all $x\in [0,1]$. Also $$\sqrt{x}-\sqrt{y}= \int_y^x \frac{1}{2\sqrt{t}} dt$$
Thus, if $(a_i,b_i) \subset [0,1],i=1,\dots,N$ are disjoint intervals, then $$\sum_{i=1}^N |\sqrt{b_i}- \sqrt {a_i}|= \int_{\bigcup_{i=1}^N (a_i,b_i)} \frac{1}{2\sqrt{t}} dt $$
Since the function $g(t)= 1/2\sqrt{t}$ is integrable, for each $\varepsilon>0$ there exists $\delta>0$ such that for all sets $A$ with measure smaller than $\delta$ we have $$\int_A g(t)dt <\varepsilon$$ Note that $A:= \bigcup_{i=1}^N (a_i,b_i)$ has measure less than $\delta$ if and only if $\sum_{i=1}^N |b_i-a_i|<\delta$ because the intervals are disjoint.
We note that the integrability of the derivative $g$ is important here. It is not true that if $f$ is absolutely continuous on every interval $[\varepsilon,1]$ and continuous at $0$, then it is absolutely continuous on $[0,1]$. One example of such a function is $f(x)=x\sin(1/x)$. This is absolutely continuous on every interval $[\varepsilon,1]$ because it has bounded derivative there, thus it is Lipschitz. However, it is not absolutely continuous on $[0,1]$, since it is not even BV there.