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?

Answer 1

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