The function $f (x) = max\{(1− x), (1+ x), 2\}$ is where $x∈(−∞, ∞)$

differentiable at all points except − 1 and 1

We draw the graph of y = 1 − x, y = 1 + x and y = 2

$f (x) = max.\{1− x, 1+ x, 2\}$

$∵f(x)=\left\{\begin{matrix}1-x,&x≤-1\\2,&-1<x<1\\1+x,&x≥1\end{matrix}\right.$

From graph it is clear that f (x) is continuous at all x and differentiable at all x except x = −1 and x = 1