In FEM, with a triangular mesh over $R^2$, could $\phi\left(x\right)=x_1\cdot\left[x\in T\right]$ be a basis function for the triangle $T$ with vertices in $\left(0,0\right), \left(0,1\right), \left(1,0\right)$? My doubts come from the fact it is not contiuous, $\phi\left(\left(1,0\right)\right)=1$ but $\phi\left(\left(1+\epsilon,0\right)\right)=0\ for\ \epsilon>0$.
Edit: basis of the trial space.
Edit: I forgot to add I want piecewise linear trial space, so the the question pretty much is about a convenient basis for it.