wcwcwc asked in 科學數學 · 1 decade ago

Advanced Calculus

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.

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  • 1 decade ago
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    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.

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