Kinky functions Are there functions whereby the left-handed and right-handed derivatives are always defined but different? What made me think about this is price elasticity of demand which for psychological reasons I think wouldn't be the same from the left and right.
 A: In Klambauer's Real Analysis pg. 101, he proves that if $f$ is a function on the open interval $(a,b)$ then there are most a countable number of points $x$ such that both $f_l'(x)$ and $f_r'(x)$ exist (including the infinite cases) but not equal.
I'll repeat his argument. Let
$$A=\{x\in(a,b): \text{both }f_l'(x)\text{ and } f_r'(x) \text{ exist, but }f_l'(x)< f_r'(x)\}$$
$$B=\{x\in(a,b): \text{both }f_l'(x)\text{ and } f_r'(x) \text{ exist, but }f_l'(x)> f_r'(x)\}$$
For each $x\in A$, chose a rational number $r_1^{(x)}$ such that $f_l'(x)<r_1^{(x)}<f_r'(x)$. After this, pick two more rational numbers $r_2^{(x)}$ and $r_3^{(x)}$ such that the following hold:
$$a<r_2^{(x)}<r^{(x)}_3<b$$
$$\text{whenever } r_2^{(x)}<y<x\text{ we can infer }\frac{f(y)-f(x)}{y-x}>r_1^{(x)}$$
$$\text{whenever } x<y<r_3^{(x)}\text{ we can infer }\frac{f(y)-f(x)}{y-x}<r_1^{(x)}$$
These inequalities imply that
\begin{equation}f(y)-f(x)< r_1^{(x)}(y-x)\end{equation}
whenever $y\neq x$ and $r_2^{(x)}<y<r_3^{(x)}$.
This process (with the Axiom of Choice) let's us construct a function $\varphi$ from $A$ into $\mathbb{Q}^3$ given by $\varphi(x)=(r_1^{(x)}, r_2^{(x)}, r_3^{(x)})$.
This function $\varphi$ is also injective. For suppose that $\varphi(x)=\varphi(y)$. This implies that
$$(r_2^{(x)}, r_3^{(x)})=(r_2^{(y)}, r_3^{(y)})\,.$$
Recall that both $x$ and $y$ are within this open interval. Thus we can infer both of these inequalities
$$f(y)-f(x)< r_1^{(x)}(y-x)$$
$$f(x)-f(y)< r_1^{(y)}(x-y)$$
Since we assumed that $r_1^{(x)}=r_1^{(y)}$, adding these inequalities gives $0<0$. Which is nonsense.
This shows that $A$ is countable. And by the same reasoning, $B$ is countable.
