Continuous Real-Valued Function/Examples/Non-Continuous Example 2
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Example of Continuous Real-Valued Function
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 $\tuple {0, 0}$.
Proof
Let $N_\epsilon$ be a neighborhood of $\tuple {0, 0}$.
The for sufficiently small $k$, the point $\tuple {k, k^2} \in N_\epsilon$.
But $\map f {k, k^2} = 1$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 24$