Continuous Real-Valued Function/Examples/Non-Continuous Example 2

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 $\map f {k, k^2} = 1$.

$\blacksquare$