prove that f is differentiable at x_0 and that f' is continuous at x_0
suppose that f is continuous on (a,b). If f is also differentiable on (a,b), except possibly at a single point where lim_x-> f'(x) exists and is finite, prove that f is differentiable at and that f' is continuous at .
suppose that f is continuous on (a,b). If f is also differentiable on (a,b), except possibly at a single point where lim_x-> f'(x) exists and is finite, prove that f is differentiable at and that f' is continuous at .
not really. basically we are asked to prove , and the conclusions will follow. to show that you can interchange the limits, probably some standard results should apply (it feels that uniform boundedness of is precisely what is needed)
not really. basically we are asked to prove , and the conclusions will follow. to show that you can interchange the limits, probably some standard results should apply (it feels that uniform boundedness of is precisely what is needed)
Oh, is that so? What makes you think what I said was incorrect?
after rereading my question over and over again, I feel like just proving L'hopital's rule is what I have to do. but what about g?
Try rereading it. The derivative exists everywhere except for maybe at that point, so L'hopital's applies and when you differentiate the numerator and denominator you get ...which...by...assumption...exists.
Try rereading it. The derivative exists everywhere except for maybe at that point, so L'hopital's applies and when you differentiate the numerator and denominator you get ...which...by...assumption...exists.
right, i understand that. But wait isn't it telling us that the limit as x->x_0 f'(x) does exist and is finite? Aye, im confusing myself, i think.