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

Question

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 09:57

score 13 · Accepted Answer · edited Dec 17 '18 at 15:46

13

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

edited Dec 17 '18 at 15:46

Community

1

answered Dec 17 '18 at 09:56

OgvRubin

1,461

1

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
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I'm not sure whether it is true or not in that case. Sorry – OgvRubin Dec 17 '18 at 10:18
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I just found the same question, see my edit! – Jiu Dec 17 '18 at 10:29
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Interesting, makes sense that I couldn’t think of a counterexample – OgvRubin Dec 17 '18 at 11:15

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

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