Wikipedia: Connectedness

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The connected (or path-connected) subsets of the real numbers R are called intervals.

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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.

A topological space is said to be connected if it cannot be divided into two disjoint nonempty open sets whose union is the entire space.

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

: { (x, y) in R² | 0 < x ≤ 1, y = sin(1/x) } U { (0, y) in R² | -1 ≤ y ≤ 1 }.

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.