Is a continuous function a continuous operator on the space of Hölder functions?

Question

Consider a continuous function $f\colon \mathbb R \to \mathbb R$ and $C^\alpha(A)$, the space of Hölder-continuous functions on a bounded set $A \subseteq \mathbb R$, and the induced function $F \colon C^\alpha(A) \to C^\alpha(A)$, $F(g) := f \circ g$.

Question: Can I, under any conditions or modifications which do not force $f$ to be an affine function, guarantee that $F$ will be continuous?

First, I directly tried to prove that $F$ is Lipschitz: $$ \Vert F(g) - F(h) \Vert_{C^\alpha(A)} \le L\Vert g - h\Vert_{C^\alpha(A)}, $$ which would, in particular, imply $$ \vert f(g(x)) - f(h(x)) - f(g(y)) + f(h(y))\vert \\ = \vert (F(g) - F(h))(x) - (F(g) - F(h))(y)\vert \\ \le L \Vert g - h\Vert_{C^\alpha(A)} \vert x - y\vert^\alpha,$$ but this seems to be not provable that because it seems impossible to split the sum into a product of $g - h$ and $x - y$.

Answer 1

This superposition operator, also known as (autonomous) Nemytskii operator, is continuous on Hölder spaces if and only if $f$ is $C^1$-smooth. This is Theorem 2 in

1. Pavel Drábek, Continuity of Nemyckij's operator in Hölder spaces, Commentationes Mathematicae Universitatis Carolinae, Vol. 16 (1975), No. 1, 37-57. Available online.

If you find this pre-TeX paper hard to read, there's a more general form of the same result in

2. Manfred Goebel, Continuity and Fréchet-differentiability of Nemytskij operators in Hölder spaces, Monatsh. Math. 113 (1992), no. 2, 107–119. MR citation.

Finally, there is a painfully detailed further study of this problem in section 7.4 of the book

3. Jürgen Appell and Petr P. Zabrejko, Nonlinear superposition operators. Cambridge Tracts in Mathematics, 95. Cambridge University Press, Cambridge, 1990. viii+311 pp. MR citation