Piecewise smooth functions..
So for i)
h(x) is continous and differentable everywhere except at x=0. For simplicity note that h(x) is an even function so we only have to test the derviative on one side.
This limit can be resolved using L'hosipitals
So now this is the form infinity divided by infinity so we take the derivative to get
So the limit is 0 and both the left and right derviative exist at that point. Also the function can be made continous if we define h(0)=0.
Now try the 2nd one.
Well, I just found that the first and fourth ones are piecewise smooth, and the second one is not piecewise smooth. But I am stuck on the third one. Since the function itself (2/(1-x)) and 0) and its derivative (2/(x-1)^2 and 0) are continuous I think it is piecewise smooth. But there is a jump at x=1. That confuses me.