# Smooth functions

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PLEASE see attachment for proper display of equations.

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!

https://brainmass.com/computer-science/algorithms/smooth-functions-equations-displayed-8619

#### Solution Preview

a.)

because,

f(n) = n*log(n), n belong to N.

for m,n belong to N and m <= n:

f(m) = m*lg(m)

and,

f(n) = n*lg(n)

because, lg(n) is slow increasing function, therefore,

f(n) >= f(m)

=> f(n) is non-decreasing function Ok.

Now,

for k,n belong to N and ...

#### Solution Summary

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

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