I'm interested in the following question:
Is the set of piecewise monotonic functions on $[a,b]$ a vector space?
I believe the answer is no, and here is my proposed counterexample:
Consider the interval $I=[0,1]$, and let $$ \begin{align} f(x)&=2x+\int_0^xt \sin \left( \frac{1}{t} \right) \mathrm{d} t \\ g(x)&=2x \end{align}$$ It's easy to see that both $f,g$ are monotonically increasing in $I$. Their difference however, has derivative $$x \sin\left( \frac{1}{x} \right), $$ which changes its sign infinitely many times in $I$.
Is this a valid counterexample? Thanks.