Exercise 7.6.7

(a)

Let $\alpha >0$ be given. Show that if $f$ is continuous at $x\in [a,b]$, then it is $\alpha$-continuous at $x$ as well. Explain how it follows that $D^{\alpha }\subseteq D$.

(b)

Show that if $f$ is not continuous at $x$, then $f$ is not $\alpha$-continuous for some $\alpha >0$. Now, explain why this guarentees that

$$D=\underset{n=1}{\overset{\infty }{\bigcup }}{D}^{{\alpha }_n}\text{ where }{\alpha }_n=1∕n$$