# Darboux Theroem

• February 1st 2010, 05:39 AM
derek walcott
Darboux Theroem
Let K be a D-domain and let f:K-->R be differentiable on the interval [a,b] as they are a subset of K. (a<b)

Let x be an element of (a,b). Show that lim (y-->x+) f'(y) and lim (y-->x-) f'(y) both exist then f' must be continuous at x.

Any help in solving this using Darboux would be greatly appreciated.
• February 1st 2010, 07:32 AM
ynj
Quote:

Originally Posted by derek walcott
Let K be a D-domain and let f:K-->R be differentiable on the interval [a,b] as they are a subset of K. (a<b)

Let x be an element of (a,b). Show that lim (y-->x+) f'(y) and lim (y-->x-) f'(y) both exist then f' must be continuous at x.

Any help in solving this using Darboux would be greatly appreciated.

According to Darboux Theorem, $f'(x)$have no discontinuity point of first kind. So $\lim_{y\rightarrow x^{-}}f'(y)=\lim_{y\rightarrow x^{+}}f'(y)=f'(x)$, so we are done.