Consider a function $f : [0, 1] \to \mathbb{R}^+$ such that $f(0) = 0$ and $f(x) \leq f(y)$ for all $ x \leq y$ (i.e $f$ is monotone). Additionally, I also restrict $f$ to be a sub-additive function i.e $f(x+y) \leq f(x) + f(y)$.
I am trying to prove that such a function need not be concave. I believe that such a function does exist but cannot find a counter example. Is there a function that satisfies my requirements?
Ideally, I would like to find a function which is also continuous and differentiable but any function not satisfying these conditions would also be fine. I believe (no formal reason) that continuous and differentiable does force the function to become concave.
Best Answer
Subadditivity is implied by a requirement that $g(x) := \frac{f(x)}{x}$ be monotonically decreasing. If $g(x) \geq g(x+y)$ and $g(y) \geq g(x+y)$, then $$f(x) + f(y) = x g(x) + y g(y) \geq x g(x+y) + y g(x+y) = f(x+y).$$ So any function $f$ that satisfies the other requirements can be subadditive as long as it does not intersect any line through the origin twice other than at the origin itself (though a single interval of coincidence with a given line through the origin, in addition to the intersection at the origin itself, is licit). This criterion is weaker than concavity (which requires that no line whatsoever intersect $f$ three times, including at the origin), and a large family of non-concave subadditive functions can be constructed as follows:
Take an increasing concave function $g: [0, 1] \to \mathbb{R}$ satisfying $g(0) = 0$.
Take some $\xi \in (0, 1)$.
Define $f(x) = \max \left\{g(x), \frac{g(\xi) x}{\xi}\right\}$. That is, $f = g$ on the interval $[0, \xi]$, and $f$ follows a continuation of the secant line from $(0, 0)$ to $(\xi, g(\xi))$ on the interval $[\xi, 1]$.
One function in this family is $$f(x) = \begin{cases} \sqrt{x} & x \leq \frac{1}{4} \\ 2x & x \geq \frac{1}{4}. \end{cases}$$
More generally, it's trivial to prove that if $f$ and $g$ are two subadditive functions, then $\max\{f, g\}$ is also subadditive, and not concave anywhere that $f$ and $g$ intersect and have different slopes.
EDIT: Since OP also asked for a smooth function, one example is $$ f(x) = 3x^3 - 6x^2 + 4x = \frac{(3x-2)^3 + 8}{9} = [3(x-1)^2 + 1]x.$$ $f$ is not concave on the whole interval $[0, 1]$ (it has an inflection point at $x = 2/3$), but it is subadditive because $f(x)/x$ is monotone decreasing on $[0, 1]$.