Let a < b, and f : ]a; b[ in R.
Assume f is continuous at x0 in ]a; b[ and f(x0) > c
for some number c in R.
Prove that there is delta > 0 such that
f(x) > c for all x in ]x0 - delta; x0 + delta[.
what would happen if, for every delta>0 there were xdelta in ]x0 - delta; x0 + delta[ with f(xdelta)=< c?
This provides an example of a proof regarding a continuous function.