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Math Help - Analysis continuity

  1. #1
    Member Jason Bourne's Avatar
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    Analysis continuity

    Let  \varphi : R \rightarrow [0,\inf) be a real valued function and assume that \varphi is continuous at 0 with \varphi(0) = 0. Let  f : R \rightarrow R be a real valued function and assume that there exists a constant c >0 such that

    \mid f(x) - f(y) \mid \leq c\varphi(x-y)

    for all x,y \in R. Prove that f is continuous.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Jason Bourne View Post
    Let  \varphi : R \rightarrow [0,\inf) be a real valued function and assume that \varphi is continuous at 0 with \varphi(0) = 0. Let  f : R \rightarrow R be a real valued function and assume that there exists a constant c >0 such that

    \mid f(x) - f(y) \mid \leq c\varphi(x-y)

    for all x,y \in R. Prove that f is continuous.
    let \varepsilon ~ > ~ 0
    let w=x-y. then
    \varphi (w) continuous at 0 \implies \exists \delta such that
    if |w - 0| < \delta \implies |\varphi(w) - \varphi(0)| = |\varphi(w)| = \varphi(w) < \frac{\varepsilon}{c}

    \implies c\varphi(w) is continuous.. (why?)

    now, \mid f(x) - f(y) \mid \leq c\varphi(x-y) = c\varphi(w) < \varepsilon. QED
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