- #1

joypav

- 151

- 0

I guess I want to construct $φ$ piece by piece of the interval to eventually cover all of [a,b]?

**Exercise:**

A continuous function $φ$ on [a, b] is called piece-wise linear provided there is a partition

$a = x_0 < x_1 < · · · < x_n = b$ of [a, b] for which φ is linear on each interval $[x_{i−1}, x_i]$.

Let f be a continuous function on [a, b] and $\epsilon$ a positive number. Show that there is a

piece-wise linear function $φ$ on [a, b] with |$f(x) − φ(x)$| $< \epsilon$ for all x ∈ [a, b].