Continuous Real-Valued Function/Examples/Non-Continuous Example 1

Example of Continuous Real-Valued Function
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 $\tuple {0, 0}$.

Proof
Let $\epsilon = \dfrac 1 2$.

Let $\delta > 0$.

Then:
 * the point $x = \tuple {\dfrac \delta 2, \dfrac \delta 2}$ satisfies $\map {d_2} {x, 0} < \delta$

but:
 * $\size {\map f x - \map f 0} = \dfrac 1 2$

Thus $f$ is continuous in each variable separately, but not in both variables jointly.