continuous function

Define $g: [0,2] \rightarrow \mathbb{R}$ as follows 

$$g(x)=f(x)-x.$$

Then 

$$g(0)=f(0) -0=1-0=1>0$$
and
$$g(2)=f(2) -2=1-2=-1<0.$$
Thus by the Intermediate Value Theorem, $g(x)$ must be zero at some point $c \in (0,2)$, i.e.
$$0=g(c)=f(c)-c  \Rightarrow f(c)=c,$$ 
for some $c \in (0,2).$