Connectedness is continuous image-closed

Statement
Suppose $$X$$ is a connected space and $$f:X \to Y$$ is a continuous map. Then, $$f(X)$$, endowed with the subspace topology from $$Y$$, is a connected space.

Weaker facts

 * Stronger than::Connectedness is coarsening-preserved

Other related facts

 * Path-connectedness is continuous image-closed

Applications

 * Intermediate-value theorem: This states that any continuous real-valued function on a connected space that takes the real values $$a$$ and $$b$$ with $$a < b$$, takes all real values in the interval $$[a,b]$$.