Definition:Discontinuous Mapping

Definition
If a function $f$ is not continuous at a point $x$, then $f$ is described as being discontinuous at $x$.

The point $x$ is called a discontinuity of $f$.

Topological Spaces
The most general definition of continuity is given in the setting of topological spaces.

Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces.

Let $f: A_1 \to A_2$ $x \in T_1$ be a mapping from $A_1$ to $A_2$.

Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $f \left({x}\right)$ there exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$.

Therefore, $f$ is discontinuous at $x$ if for some neighbourhood $N$ of $f \left({x}\right)$ and every neighbourhood $M$ of $x$, $f \left({M}\right) \nsubseteq N$.