notation: mapping $f\rightarrow\int_{0}^{1}f$ of $C(I)$ to $\mathbb{R}$

Question

Can anyone explain what the following notation refers to:

mapping $f\rightarrow\int_{0}^{1}f$ of $C(I)$ to $\mathbb{R}$

This is from a question asking about uniform continuity of $f$ on $C(I)$, but I am not quite clear on the above notation

Answer 1

You have spelled out the definition already: It refers to the mapping

$$T: C(I) \to \mathbb R, $$

which is defined by $T(f) = \int_0^1 f$.

(I am assuming $[0,1] \subset I$)

Answer 2

A mapping $L: A \to B$ works in the following way:

Given an element $x \in A$, it is mapped to a unique element $y = L(x) \in B$.

Now you mapping you asked works in the following way:

Given a continuous function $f$ on the interval $I$, it is mapped to a unique number $\int_0^1 f \, dx \in \mathbb{R}$.

Mathematically we write $$f \in C(I) \mapsto \int_0^1 f \, dx \in \mathbb{R}.$$

As John's answer has said, we usually denote such a mapping by $$L:C(I) \to \mathbb{R} \qquad L(f) = \int_0^1 f \, dx, \, \text{ for } f \in C(I).$$