Sum of convex function and increasing function

Question

I have a sum of increasing function and a convex function over some domain. Can I say that the sum is also a convex function ? Or when can i say that sum of convex function with increasing function is convex and sum of convcave function and increasing function is concave? Functions are continous... Thanks a lot for helping...

score 1 · Answer 1 · answered Aug 09 '20 at 18:10

1

No. Let $f(x) = x^2$ and $g(x) = \begin{cases}1 & x>0\\ 0 & x\leq 0\end{cases}$.

Then $f$ is convex, $g$ is monotone increasing, but $f+g$ is not continuous on the interior of its domain and so cannot be convex.

answered Aug 09 '20 at 18:10

Jürgen Sukumaran

7,645

It doesn't matter. And if you want to ask a question about continuous functions, state that in your question. You didn't say anything about continuity, that was your question. – Jürgen Sukumaran Aug 09 '20 at 18:14

score 1 · Accepted Answer · answered Aug 09 '20 at 18:11

1

The first statement is incorrect, consider the convex function $x^4$ and the increasing function $x^3$ whose sum is not convex (consider $x = 0$ and $x = -\frac{3}{4}$).

answered Aug 09 '20 at 18:11

ViktorStein

4,838

1

$|x| + (-e^{-x})$. – Jürgen Sukumaran Aug 09 '20 at 18:19

Sum of convex function and increasing function

2 Answers2