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Math Help - Symmetrically Continuous

  1. #1
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    Symmetrically Continuous

    A function f is said to be symmetrically continuous at x_0 if

    \lim_{h{\rightarrow{0^+}}} [f (x_0 + h) - f (x_0 - h)] = 0.

    Show that if f is continuous at x_0, it is symmetrically continuous there but not the conversely.
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  2. #2
    Senior Member Tinyboss's Avatar
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    The forward direction is trivial--just use the fact that the limit of the difference is the difference of the limits, together with the limit definition of continuity.

    Disproving the converse is almost as easy--just consider

    f(x)=\begin{cases}0&x\ne x_0\\1&x=x_0.\end{cases}.
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