Assume g:(a,b)->R is differentiable at some point c belong to (a,b). If g'(c)not= 0 show that there exists a delta neighborhood V_delta (c) subset or equal to (a,b) for which g(x) not= g(c) for all x belong to V_delta (c).
Since g'(c) is not 0, without loss of generality, we assume g'(c)>0. We know g'(c) is the limit of (g(x)-g(c))/(x-c) as x->c. When ...
A neighborhood is investigated.