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    Smooth functions

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    Let f be a function from N to R+ . Function f is eventually nondecreasing if
    such that . Function f is smooth if it is eventually
    nondecreasing and such that .
    (a) Is the function + such that a smooth function? Verify!
    (b) Is the function + such that = a smooth function? Verify!

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    Solution Preview

    f(n) = n*log(n), n belong to N.
    for m,n belong to N and m <= n:
    f(m) = m*lg(m)
    f(n) = n*lg(n)
    because, lg(n) is slow increasing function, therefore,
    f(n) >= f(m)
    => f(n) is non-decreasing function Ok.
    for k,n belong to N and ...

    Solution Summary

    Smooth functions are analyzed. Increasing and nondecreasing functions are provided.