If a function is differentiable, does that mean its higher order derivatives are also differentiable?
Oh, that's not true! A polynomial is infinitely differentiable. I can not imagine why you would think it will "eventually have a removable discontinuity at x= 0". The derivative of any polynomial is eventually identically 0 and that's a very differentiable function!
Instead, to find an example of a differentiable function that is not infinitely differentiable, start with a discontinuous function and integrate it.
For example, integrating f(x)= -1 if x< 0 and 1 if , which is not continuous at x=0, gives g(x)= -x if x< 0 and g(x)= x if (I have chosen the constant of integration to be 0). In other words, g(x)= |x| which is continuous for all x but not differentiable at x= 0. Integrating again gives if x< 0 and if . That function is differentiable at x= 0 but not twice differentiable. Integrating again would give a function that is twice differentiable but not three times differentiable.