Discontinuity (Real Analysis)/Examples

Examples of Discontinuities in the context of Real Analysis

Example 1

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \dfrac 1 {1 - x}$

Then $f$ has a discontinuity at $x = 1$, as $f$ is not defined there.

Example 2

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} \dfrac 1 k & : 0 \le x \le k \\ 0 & : \text {otherwise} \end {cases}$

Then $f$ has a discontinuity at $x = 0$ and $x = k$

Example 3

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \dfrac 1 {x^2 - 4}$

Then $f$ has a discontinuity at $x = -2$ and $x = 2$.

These are infinite discontinuities.