Mathematics

A piecewise function can fail to be differentiable at the boundary points where the formula changes, even if it's continuous there. The key is to check whether the left and right derivatives match at each transition point—if they don't, there's a sharp corner or cusp where the function isn't smooth. You'll need to compute one-sided derivatives at $x = 1$ and $x = 3$ and compare them.

A piecewise function can fail to be differentiable at the boundary points where the formula changes, even if it's continuous there. The key is to check whether the left and right derivatives match at each transition point—if they don't, there's a sharp corner or cusp where the function isn't smooth. You'll need to compute one-sided derivatives at $x = 1$ and $x = 3$ and compare them.