TITLE: What is the difference between a function and a curve?
QUESTION [8 upvotes]: Are all curves functions? To my current knowledge a function gives only one output for a given input so x->f(x) , but for example a circle which is a curve is "made" out of two functions, so generally speaking a circle is a curve yet not a function, so can someone provide me a good distinction between them and good definition of a curve. Thanks!

REPLY [8 votes]: I think that the most general definition of ''curve''  that correspond to our intuition is:

A curve is a continuous function $\gamma: I \to X$ where $ I \subset \mathbb{R}$ is an interval and $X$ is a topological space. 

So, every curve is a function, but this does not means that, If $X= \mathbb{R}^2$ than any curve can be expressed as a function $f:\mathbb{R} \to \mathbb{R} \qquad y=f(x)$.
In this case, as you notice, a circle is a curve, but we have not a single function $f:\mathbb{R} \to \mathbb{R} $ such that the points of the circle are the graph of $f$.
But note that these points $P=(x,y)$ on the circle can be represented as the domain of a function $f:[0,2\pi) \to \mathbb{R}^2$ as  $f(t)=(\cos t,\sin t)$.