Let [t] denote the greatest integer t.

Question

Let $[t]$ denote the greatest integer $\le t$. Let $f(x)=x-[x]$, $g(x)=1-x+[x]$, and $h(x)=\min\{f(x),g(x)\}$, $x\in[-2,2]$. Then $h$ is

Accepted Answer

$f(x)=\{x\}$, $g(x)=1-\{x\}$. $h(x)=\min\{\{x\},1-\{x\}\}$ is continuous everywhere (fractional part functions are continuous except at integers, but the min structure removes the jump), while it has corners at $x=-\frac32,-1,-\frac12,0,\frac12,1,\frac32$ in $(-2,2)$ --- more than four points, so $h$ is continuous in $[-2,2]$ but not differentiable at more than four points.

Solutions · 1