Continuous vector field on surface On plane we can easy to define continuous vector field:
$$\mathbb{R}^2\ni x\longmapsto v(x)\in\mathbb{R}^2,$$
where real-valued function $v_1(x)$ and $v_2(x)$ are continuous.
But when we try to define field on arbitrary 2-dimensional surface we have a trouble:
$$v(x_1)\in T_{x_1}S,~~~v(x_2)\in T_{x_2}S,$$
and for different points $x_1$ and $x_2$ tangent spaces $T_{x_1}S$ and $T_{x_2}S$ are different. So function $v_1$ and $v_2$ no sense.
Question: how to define continuous vector field on surface?
Thanks.
 A: Of course, it'd be nice if all tangent vectors to $M$ were elements of some big space; then we could talk about the function assigning a point to the vector at that point being "continuous", 
or "smooth", etc. If we were working in a submanifold of $\mathbb{R}^n$, perhaps we could try to compare the tangent vectors within $\mathbb{R}^n$; but when we're working in an abstract manifold, we have to do everything ourselves.
The answer is simply to "put all of the $T_pM$'s together", and voila, we have a big space that contains all of our tangent vectors! 
The resulting object is known as the tangent bundle, and as far as I know, this is the main reason why it was invented. As a set, it's precisely
$$TM=\coprod_{p\in M}T_pM$$
and it also gets a topology (though clearly it's not going to be the disjoint union topology), and it also has a natural smooth manifold structure. It comes with  a natural projection map $\pi:TM\to M$ defined by taking the elements of $T_pM$ to $p$. 
A continuous vector field is then just a continuous section of $\pi$. 
