Class of functions that satisfy $\frac{f(x+th)-f(x)}{t^{\alpha}} \to d \text{ as $t \to 0$}$

Question

Suppose $f:X \to Y$ is a function between Hilbert spaces that satisfies, given $x, h \in X$, $$\frac{f(x+th)-f(x)}{t^{\alpha}} \to d \text{ as $t \to 0$}$$ for some $d \in Y$. Is there a name for such functions and is it a useful class? Is there such a thing as "directionally Holder continuous functions with exponent $\alpha$"?

Answer 1

There would not be a name for this class, and you are not likely to find examples of such functions with nonzero limits $d$ everywhere. For example, if $f:[0,1]\to\mathbb{R}$ satisfies the given condition everywhere with $d=0$, then it must be nowhere differentiable, hence wildly oscillating (a Weierstrass-type function) -- and this does not mesh well with the assumption of $\frac{f(x+th)-f(x)}{t^{\alpha}}$ having a limit.

However, there is a reasonable class of functions that satisfy the stated condition with $d=0$. Sometimes denoted $c^{0, \alpha}$, it's defined as the closure of the set of smooth functions in the usual Hölder space $C^{0,\alpha}$. Such functions satisfy $f(x)-f(y) = o(\|x-y\|^\alpha)$ as $x - y\to 0$, which is a stronger (undirectional and uniform) version of the condition you have.

Is there such a thing as "directionally Holder continuous functions with exponent $α$"?

It's a meaningful term, but it does not match the definition you stated. Hölder continuity does not require the existence of any such limit.