If f : R → R is a concave function such that lim x → ∞ ( f ( x ) − f ( x − 1 ) ) = 0 then f increasing.

Question

If   f  :      R    →      R   is a concave function such that       lim          x      →      ∞        (  f  (  x  )  −  f  (  x  −  1  )  )  =  0 then   f increasing.

Tasinazzokbc · Accepted Answer

Hint. Since   f is concave it follows that for any       x    0    ∈      R   the function      x    0    ∈      R  is decreasing in       R    ∖  {      x    0    }. Here it is a reference for the convex function   −  f.Now assume that   f is not increasing. Then there are x0,x1 such that       x    0    ,      x    1   and   f  (      x    1    )  &amp;lt;  f  (      x    0    ). It follows that  0  &amp;gt;  R  (      x    1    ,      x    0    )  ≥  R  (      x    0    +  k  +  1  ,      x    0    )  ≥  R  (      x    0    +  k  +  1  ,      x    0    +  k  )where   k is any positive integer such that       x    0    +  k  &amp;gt;      x    1  .

If f:R->R is a concave function such that limx_(->oo)(f(x)−f(x−1))=0 then f increasing.

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