Let us define the function as we have |x| here ..
So
f(x) = 1 / (1 + x) , if x >= 0
= 1 / (1 - x) , if x <= 0
For continuity , we check at x = 0 only , so
LHL = RHL = f(0) = 1
f' (x) = - 1 / (1 + x)^{2} , if x >= 0
= -1 / (1 - x)^{2} * (-1) ,
= 1 / (1 - x)^{2 } , if x <= 0
So Left derivative at x = 0 : 1
Right derivative at x = 0 : -1
So as Left derivative != Right derivative here
Hence f(x) is not differentiable at x = 0
Hence the given function is continuous but not differentiable due to x = 0..