Definition:Discontinuous Mapping/Real Function

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Let $f$ be a real function.

Then $f$ is discontinuous if and only if there exists at least one $a \in \R$ at which $f$ is discontinuous.

At a Point

Let $A \subseteq \R$ be a subset of the real numbers.

Let $f : A \to \R$ be a real function.

Let $a\in A$.

Then $f$ is discontinuous at $a$ if and only if $f$ is not continuous at $a$.