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I am a 1st year-2nd semester student of the department of Applied Mathematics @ NTUA. Some days ago, I had a really interesting conversation with one of my old school mathematics teacher about a theorem for concave up (or down) functions which is boldly stated in the book of the final grade of high-school, which is also studied for the panhellenic exams.

The definition of a concave up function, is stated as :

Let $f$ be a continuous function over $D \subseteq \mathbb R $ and also differentiable in the interval of $D$. We say that : The function $f$ is concave up, if the derivative of $f$,$f'$ is strictly increasing in the interval of $D$.

According to that definition, the answer to the following question is TRUE : If $f$ is a concave up function over $D \subseteq \mathbb R $ and $1$,$2$ belong in the interval of D, then $f'(1) < f'(2)$.

The formal definition for a concave up/convex function $f:X \to \mathbb R$ though, is : $ f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2)$ $\forall x_1,x_2 \in X, \forall t \in [0,1] $.

Now, my question is : Is the book's definition perfectly solid, or is it possible to find a function which is concave up BUT NOT diffentiable ? All I could think off were function that you cannot integrate in the set of standard mathematical equations, but that doesn't mean that by definition they aren't concave up (e.g. $f(x) = \int \int e^{-x^2}dx)$

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Try $f(x)=|x|$ on $\mathbb R$. Then, for all $t\in[0,1]$ and $x_1,x_2\in\mathbb R$, \begin{align*}f(tx_1+(1-t)x_2)&=|tx_1+(1-t)x_2|\leq |tx_1|+|(1-t))x_2|=t|x_1|+(1-t)|x_2|\\ &=tf(x_1)+(1-t)f(x_2), \end{align*} therefore $f$ is convex.

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  • $\begingroup$ Not sure if convex means concave up (I do not know the exact English terminology). For example $f(x) = x^2$ is concave up. Excuse me if by any chance my terminology wasn't correct. Anyway if it was correct, then we consider $f(x) = |x|$ concave up ? $\endgroup$ – Rebellos May 20 '16 at 22:59
  • $\begingroup$ "Convex" is the term for "concave up" used outside of elementary calculus. $\endgroup$ – Robert Israel May 20 '16 at 23:00
  • $\begingroup$ Good, so I was correct. Now, straight to the point, we consider $f(x) = |x|$ a concave up function, so the general definition of the school book falls over, which means it only studies differentiable functions. $\endgroup$ – Rebellos May 20 '16 at 23:02

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