Prove or Disprove: Summation of two functions (at least one discontinuous) supports IVP, if both of them support IVP.

Question

Let, f and g be two functions on R, support IVP and at least one them is discontinuous. Then prove or disprove ( with example) whether f+g also supports IVP.

If f and g, both are continuous, then it is easy to say that, f+g supports intermediate-value-property. But here, we have to show, whether this result holds for discontinuous function or not.

Answer 1

Consider the interval $I=[0,1]$.

$f(x)=x$ is continuous on $I$. The map $$h(x)=\begin{cases} 0 & \mbox{for} &0 \le x <1/2\\ 1 & \mbox{for} &1/2 \le x \le 1 \end{cases}$$

Doesn't support the IVP. However $g=h-f$ which is discontinuous does.

Unfortunately my answer is wrong...

Answer 2

For any $s \in [-1,1]$, one can show that $f_s : \mathbb R \to \mathbb R$ defined via $$f_s(x):= \begin{cases} \sin(1/x) & \text{if } x \ne 0 \\ s & \text{else}\end{cases}$$ satisfies the IVP.

However, the difference of two such functions does clearly violate the IVP.