Marie and Fred are arguing again, this time about the function:
Fred says that if x > 0 then g(x) is a non-constant function whose derivative is zero. Marie says that's nonsense, and even if it weren't, for x < 0, g(x) is a smooth function whose derivative is never zero.
Please help them resolve this dispute.
Thanks to ThePerfectHacker for the repy. Yes you are basically right. I'm not sure what you mean by "those two vertical lines" though. The graph of y=g(x) looks like this:
What seems strange to me is that a function defined by a nice single line analytic formula, namely
Should be a step function, i.e. constant but for a jump discontinuity for x>0 but a smooth monotonically increasing function for x<0. The explanation comes from the fact that if you let
then using, for example the double angle formula for the tangent function, it is not hard to show that . This, however, can happen in different ways, and for x<0 it happens via, , whereas for 0<x<1 it happens as and after the discontinuity at x=1 it happens as .