suppose f is a function defined on an open set S,
S as a subset of R^n.
show that if the partial derivatives Djf exist and are bounded on S, then f is continuous on S.
- mathmanliuLv 71 decade agoFavorite Answer
(Prove that f is conti. at x)
S open, so there exists δ1>0 such that Ball(x,δ1) contained in S.
Let M>0 be the max. L^2 norm of grad(f) on S.
For ε>0, takeing δ= min(δ1, ε/M), so
if y in Ball(x, δ), then
| f(y)-f(x)|= |grad(f)(z)|‧|y-x| (mean value thm.)
<= M|y-x| (Cauchy inequaity)
< M*ε/M =ε
ie. f is conti at x.