Question

Determine whether y is differentiable at x=0 . If it is differentiable, find the derivative at x=0 .
y = {x if x<= 0, x+1 if x>0}


y' = ( 1 if x<= 0, 1 if x>0)
Yes , it is differentiable at 0. Is the derivative at x=0, 1?

Answer 1

while y'=1 at x=0, y(x) is discontinuous there.
y(0) -> 0 on the left, y(0) = 1 on the right.
Do the graph, and you can see that y is not differentiable at x=0
read the definition again.