Limits, Differentiability, Continuity

[mscbuck]

Homework Statement

Suppose that f is differentiable in some interval containing "a", but that [tex]f'[/tex] is discontinuous at a.

a.) The one-sided limits lim f'(x) as x[tex]\rightarrow[/tex] a+ and lim f'(x) as x[tex]\rightarrow[/tex]a- cannot both exist

b.)These one-sided limits cannot both exist even in the sense of being +Inf or -Inf

Homework Equations

The Attempt at a Solution

For a.), I think I shall begin trying to take a manipulation of Theorem 7 in our book, which states that if f is continuous at a, and that f'(x) exists for all x in some interval containing a (except for perhaps at x - a), and if the lim f'(x) as x--> a exists, then f'(a) exists and f'(a) = lim f'(x) as x-->a

Is that the right place to start looking for proving part a.)? It seems that it's saying many of the same things, except our problem states that f is NOT continuous at a, but is differnetiable in some interval that contains A. I see what it says just reading it, but having some trouble putting it down onto paper

For part b.) I was told to use Darboux's Theorem, but am having trouble figuring out what it can say to help me prove part B.

Thanks!

 

[mscbuck]

Would a proof by contradiction work better? Could we suppose that f' is indeed continuous at a hopefully to find something? I'm trying right now but not really getting anywhere :/