Q:

Non trivial projector on $C^{0}( \mathbb{R}, \mathbb {R} )$?

I am looking for a projection operator $P: C^{0}( \mathbb{R}, \mathbb {R} ) \rightarrow C^{0}(\mathbb{R}, \mathbb {R})$, i.e. a linear operator s.t. $T^2 = T$, continuous or not.
Edit :
I found a lot projectors :
For examples :
The composition with map $g_{a,b}$ such $g|_{ [a,b]}=Id$ and $g(x)=a \ \forall x \leq a$ and $g(x)=b \ \forall x \geq b$
The "constant" operator $T_y:f \rightarrow (x \rightarrow f(y))$
The "even" operator : $f \rightarrow (x \rightarrow \frac{f(x)+f(-x)}{2}$
The "odd" operator : $f \rightarrow (x \rightarrow \frac{f(x)-f(-x)}{2}$
If we have a projector $T'$ on $[a,b]$ and $T''$ on $[c,d]$ and $b < c$, we can build $T''$ on $\mathbb{R}$ :
$T(f)=T'(f)$ on $[a,b]$
$T(f)=T''(f)$ on $[c,d]$
$T(f)(x)=T'(f)(a) on $(- \infty,a]$
$T(f)(x)=T''(f)(a) on $[d, \infty $
$T(f)(x)=(x-b) \frac{T(f)(c)-T(f)(b)}{c-b} + f(b)$ on $[b,c]$

A:

Example: let $P: C^{0}( \mathbb{R}, \mathbb {R} ) \rightarrow C^{0}(\mathbb{R}, \mathbb {R})$ be definde by $P(f):=f-f(0).$
$P$ is linear and $P^2=P.$

