Is every real valued function on an interval a sum of two functions with Intermediate Value Property?

If $I$ is an interval of real numbers , then is it true that any function $f:I \to \mathbb R$ can be written as $f=f_1+f_2$ , where $f_1 , f_2 : I \to \mathbb R$ have the Intermediate value property?

Yes, it is true. Partition $I$ in two sets $A,I\setminus A$, both having cardinality $c$ and with the property that every interval in $I$ intersects both $A$ and $I\setminus A$. Let $g,h$ be two functions which take on every value in every relative subinterval of $A$ (and $I\setminus A$ respectively). We can extend $g$ to $I\setminus A$ and $h$ to $A$ in such a way that $g+h=f$ (by defining $g$ as $f-h$ on $I\setminus A$ and $h$ as $f-g$ on $A$) holds over the whole $I$. Since $g$ and $h$ obviously have the Darboux property, we are done.