9
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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) $\endgroup$ – Yanko Dec 17 '18 at 9:57
13
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question? $\endgroup$ – Jiu Dec 17 '18 at 10:05
  • 1
    $\begingroup$ I'm not sure whether it is true or not in that case. Sorry $\endgroup$ – Olof Rubin Dec 17 '18 at 10:18
  • 2
    $\begingroup$ I just found the same question, see my edit! $\endgroup$ – Jiu Dec 17 '18 at 10:29
  • 1
    $\begingroup$ Interesting, makes sense that I couldn’t think of a counterexample $\endgroup$ – Olof Rubin Dec 17 '18 at 11:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.