Exercise 6.7.2

Recall Theorem 4.4.7, which states that a continuous functions over a compact set is uniformly continuous over that set. Given $𝜖>0$, apply uniform continuity on $f$ with $𝜖∕2$ to obtain some $\delta >0$, and partition $[a,b]$ into uniform segments, with each segment length lower than $\delta$. Define $\varphi (x)$ at the endpoints of each segment to be equal to $f(x)$, and to linearly interpolate between segment endpoints.

For any $x\in (a,b)$, let $q$ be the largest segment endpoint less than $x$, and $r$ be the following segment endpoint. (If $x=a$ or $x=b$ then these aren’t necessarilly defined, but then $\varphi (x)=f(x)$ so there’s nothing to worry about.) Since $\left|x-q\right|<\delta$ we have that $\left|f(x)-\varphi (q)\right|<𝜖∕2$. We similarly also have $\left|\varphi (q)-\varphi (r)\right|<𝜖∕2$. Also, note that $\varphi (x)$ must lie between $\varphi (q)$ and $\varphi (r)$, so $\left|\varphi (q)-\varphi (x)\right|\le \left|\varphi (q)-\varphi (r)\right|<𝜖∕2$. Applying the triangle inequality leaves us with $\left|f(x)-\varphi (x)\right|<𝜖$ as desired.