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.

Rating

註:"‧"為內積符號

(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.