### *41 … is any continuous function with

Explain the difference between a continuous function and a continuable function. Are continuous functions always continuable?

### *41
$f(x)$ is any continuous function with $f(0)=f(1)$.
(a) Draw a typical $f(x)$. Mark where $f(x)=f(x+\frac{1}{2})$.
(b) Explain why $g(x)=f(x+\frac{1}{2})-f(x)$ has $g(\frac{1}{2})=-g(0)$.
(c) Deduce from (b) that (a) is always possible: There _must_ be a point where $g(x)=0$ and $f(x)=f(x+\frac{1}{2})$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §2.7 Continuous Functions, #40