Space of continuous functions

The set of all real-valued continuous functions on a given topological space.

Space of continuous functions

The space of continuous functions on a topological space $X$ is

C(X)=\{f:X\to\mathbb{R} \mid f \text{ is continuous}\},

where continuity is in the sense of a continuous map . It is naturally a vector space under pointwise addition and scalar multiplication.

On domains where continuous functions are bounded (for example, on a compact interval), $C(X)$ can be equipped with the supremum norm and the associated uniform metric , linking function-space topology to uniform convergence . Theorems such as Arzelà–Ascoli and Stone–Weierstrass are statements about subsets of $C(X)$ .

Examples:

On $[0,1]$ , every polynomial function belongs to $C([0,1])$ .
On $[-1,1]$ , the function $f(x)=|x|$ belongs to $C([-1,1])$ .