what is this sort of function called? I am doing an assignment but I do not know how to do this problem. I have the following:
$$
f(x)=
\begin{cases}
0 & \text{for $x<0$},\\
x & \text{for $x\geq 0$}.
\end{cases}
$$
We are meant to determine whether it is continuous or not. I do not know why there is like a bracket with a $0$ up top and an $x$ at the bottom, nor do I understand what is meant by for $x \geq 0$. Basically, I do not understand the whole thing. What is this sort of function called so I can do some research on it (I plan to look it up on khan academy)?  

As others have mentioned, what you have is called a piecewise function. This literally means "pieces" of your function are defined differently. 
For 
$$
f(x)=
\begin{cases}
0 & \text{for $x<0$},\\
x & \text{for $x\geq 0$},
\end{cases}
$$
it may help you to actually draw what the function $f(x)$ looks like for some sample values. For example, the following is what your $f(x)$ looks like on the interval $-5\leq x\leq 5, 0\leq y\leq 5$:
$\color{white}{Please do center this}$
Added: It may help (for the OP) to know that the following block of code (Mathematica) was used to generate the figure above: Plot[Piecewise[{{0, x < 0}, {x, x >= 0}}], {x, -5, 5}]
You can see the effect on Wolfram|Alpha if you like (and you can toy around with the parameters to see how things might change if you so desire).
