Continuous Real-Valued Function/Examples
Jump to navigation
Jump to search
Examples of Continuous Real-Valued Functions
Non-Continuous Example 1
Let $f: \R^2 \to \R$ be the real $2$-variable function defined as:
- $\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = \begin {cases} 0 & : \tuple {x_1, x_2} = \tuple {0, 0} \\ \dfrac {x_1 x_2} {x_1^2 + x_2^2} & : \text {otherwise} \end {cases}$
Then the restrictions of $f$:
- $f_{\restriction \R \times \set 0}$
- $f_{\restriction \set 0 \times \R}$
are both constant functions with value $0$ for all arguments.
Hence both are continuous at $\tuple {0, 0}$.
But $f$ is not continuous at $\tuple {0, 0}$.
Non-Continuous Example 2
Let $f: \R^2 \to \R$ be the real $2$-variable function defined as:
- $\forall \tuple {x, y} \in \R^2: \map f {x, y} = \begin {cases} 0 & : y = 0 \\ \dfrac {x^2} y \end {cases}$
Then the restrictions of $f$:
- $f_{\restriction \tuple {x, y} \in \R^2: y = m x}$
is continuous.
But $f$ is not continuous at $\tuple {0, 0}$.