**Artur Wachowicz**

## Preimages of Residual Sets of
Continuous Functions under Operation of Superposition

**Abstract:**

Let $C=C[0,1]$ denote the Banach space of continuous real functions on $[0,1] $
with the $\sup $ norm and let $C^{\ast }$ denote the topological subspace of $C$
consisting of functions with values in $[0,1].$ We investigate the preimages of
residual sets in $C$ under the operation of superposition $\Phi :C\times C^{\ast
}\rightarrow C$, $\Phi(f,g)=f\circ g$. Their behaviour can be different. We show
examples when the preimages of residual sets are nonresidual of second category,
or even nowhere dense, and examples when the preimages of nontrivial residual
sets are residual.

**Keywords:**

Baire category, residual set, space of continuous functions, superposition.

**MSC 2000:** 26A15, 54E52