The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f : [0, 1] -> X with f(0) = x and f(1) = y.
Every path-connected space is connected. An example of a connected space that is not path-connected is the topologist's sine curve. This is the compact plane set
However, subsets of R are connected if and only if they are path-connected. These subsets are the intervals of R.
If X and Y are topological spaces, f : X -> Y is continuous, and X is connected (respectively, path-connected), then f(X) is connected (respectively, path-connected). The intermediate value theorem can be considered as a special case of this result.