# Does the sum of two functions satisfying the intermediate value property also have this property?

If functions $$f$$ and $$g$$ both satisfy the intermediate value property, does their sum also satisfy this property? If not, what if I suppose in addition that $$f$$ is continuous?

Thanks in advance!

Edit: I found the second part of my question here: Is the sum of a Darboux function and a continuous function Darboux?

• The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order) – Yanko Dec 17 '18 at 9:57

## 1 Answer

Consider the functions, $$f:[0,1]\rightarrow [-1,1]$$ and $$g:[0,1]\rightarrow[-1,1]$$ where

$$f = \begin{cases}\sin\frac{1}{x},& x>0 \\ 0, & x = 0\end{cases}$$ and $$g = \begin{cases}-\sin\frac{1}{x},& x>0 \\ 1, & x = 0\end{cases}$$

• Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question? – Jiu Dec 17 '18 at 10:05
• I'm not sure whether it is true or not in that case. Sorry – Olof Rubin Dec 17 '18 at 10:18
• I just found the same question, see my edit! – Jiu Dec 17 '18 at 10:29
• Interesting, makes sense that I couldn’t think of a counterexample – Olof Rubin Dec 17 '18 at 11:15