Are curves in a level set continuous? Wikipedia defines a level set as 

a level set of a real-valued function of $n$ real variables $f$ is a set of the form $$L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}$$

I often seen level sets drawn as curves in $\Bbb{R}^2$ like this

My Question:
Why do we assume a level set is continuous? Why couldn't it be just a collection of random points all with the same value instead of curves? 
 A: If $f$ is continuous, then the level set $\{f=c\}$ has to be closed. That's about all you can say: Given an arbitrary closed subset $E$ of $\mathbb {R}^n,$ there is $f\in C^\infty(\mathbb {R}^n)$ such that $\{f=0\} =E.$ To get nice smooth things as level sets, there is usually an added assumption hanging around, such as $\nabla f\ne 0$ on $E.$
A: If you have a differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}$, and want to look at $A=f^{-1}(\{c\})$ for some $c \in \mathbb{R}$, you can say $A$ is an embedded $n-1$ manifold in $\mathbb{R^n}$, provided that the gradient is not zero in any point of $A$. This follows from the inverse function theorem. 
In your image, you are looking at the level set of a function $f:\mathbb{R^2} \rightarrow \mathbb{R}$. Therefore, if all points are "well-behaved" on the pre-image of a given value, you will have a curve as a level set (a closed loop or a curve that goes to infinity, by a characterization of 1-manifolds). 
Notice that this is not always the case! Take $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x,y)=xy$ for instance. Then $f^{-1}(\{0\})$ is the union of the two axis, which is not a 1-manifold.
A: In general, yes, the level set is just a bunch of points. But for most functions you will see, the level set is a continuous loop. It doesn't have to be a loop or line, for instance the level set of the zero function at zero is the entire domain, which could be the entire plane. The reason that the picture you gave is the usual visualization for a level set (of a function with a 2D domain) is that it commonly arises that way from a 'well-behaved" function, for instance one that is continuous. The set of level curves is then a nice way of visualizing the gradient function or topographical maps, which only work for continuous multi-variable functions.
