3. In the coordinate plane, there are two regions $M$ and $N, M$ is defined by $\left\{\begin{array}{l}y \geqslant 0, \\ y \leqslant x, \quad N \text { is a region that varies with } t, \\ y \leqslant 2-x,\end{array}\right.$ it is determined by the inequality $t \leqslant x \leqslant t+1$, where the range of $t$ is $0 \leqslant t \leqslant 1$, then the common area of $M$ and $N$ is the function $f(t)=$ $\qquad$
Given $E(1,1), F(2,0)$. Note that $\triangle O A B$ and $\triangle D C F$ are isosceles right triangles, and $|O B|=t,|C F|=2-(t+1)$ $=1-t$, thus $S_{\triangle O A B}+S_{\triangle D C F}=\frac{1}{2}\left[t^{2}+(1-t)^{2}\right]$ $=\frac{1}{2}\left(2 t^{2}-2 t+1\right)$.
Therefore, $S_{A B C D E}=S_{\triangle O E F}-\left(S_{\triangle O A B}+S_{\triangle D C F}\right)$ $=1-\frac{1}{2}\left(2 t^{2}-2 t+1\right)=-t^{2}+t+\frac{1}{2}$.