# Thread: Darboux Theroem

1. ## 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.

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