Sum of Two functions with IVP I am looking for a different   example  from one below   of two functions that satisfy intermediate value property (IVP) but their sum  not. Here is my work
the Let $f(x)=g(x)=\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$, $g(0)=1$. Then,  both of $f$ and $g$ are Darboux functions but $f-g$ is not. $f$ and $g$ satisfy IVP,. To see $f-g$ does not satisfy IVP
Notice that $(f-g)(x)=0$ for all $x\neq 0$ and $(f-g)(0)=-1$. Now, $(f-g)(0)<-\frac{1}{2}<(f-g)(1)$ but there is no $x\in(0,1)$ such that $(f-g)(x)=-\frac{1}{2}.$
I know this is right but  I would like to see another example.
 A: Ah!  Someone interested in functions with the intermediate value property.
These animals have their own name which makes it easier to search for them.  They dislike the acronym "IVP."  We call them Darboux functions, named after the French mathematician Jean-Gaston Darboux  (1842-1917).  He proved that all derivatives had the IVP property and he showed that there were derivatives that were quite discontinuous, thus killing any suggestion that the IVP property is equivalent to continuity.
Here are some theorems that you will certainly find entertaining in your quest to meet some more of these curious creatures.  The first theorem shows, in a rather more surprising way than you might expect, that the sum of two Darboux functions need not be Darboux.
Theorem 1.   Let $f:\mathbb R\to \mathbb R$ be a completely arbitrary function.  Then there exist two Darboux functions $g$ and $h$ such that $f=g+h$.
Theorem 2.  Let $f:\mathbb R \to \mathbb R$ be a completely arbitrary function.  Then there exist a sequence $\{g_n\}$ of Darboux functions that converges pointwise to $f$.
Theorem 3.  Let $f:\mathbb R \to \mathbb R$ be a completely arbitrary function. Then there  exists a Darboux function  $g$   such that the set of points
$$ \{x\in \mathbb R: f(x)\not = g(x) \}$$
is both measure zero and first category.
These three theorems are proved in Chapter 1 of  Andy Bruckner's monograph [1] cited below.  He adds this comment:

A good  deal more can be said about Darboux functions.  But we shall
be concerned with derivatives and related classes of functions, and,
since these classes of functions are generally contained in Baire
class 1, further discussions of Darboux functions would really be
peripheral to our needs.  The main purpose in the present chapter was
to provide a bit of the flavor  of Darboux functions.  We refer anyone
interested in further pursuing the   subject to [this] expository
article
A. Bruckner and J. Ceder,  Darboux continuity, Jber. Deutsch Math. Ver. 67, (1965), 93-117.

Jack Ceder passed away some time ago, but Andy is still with us and still knows more about Darboux functions than you might care to learn.
REFERENCE:
[1] https://www.amazon.com/Differentiation-Real-Functions-Crm-Monograph/dp/0821869906
