$g(x,y) = xy/(x^{2}+y^{2})$, extended by $g(0,0)=0$, is continuous in each variable separately and not continuous at the origin

Refuted claim: a function $g : \mathbb{R}^{2} \to \mathbb{R}$ that is continuous in each variable separately — that is, for which $t \mapsto g(t,b)$ and $t \mapsto g(a,t)$ are continuous on $\mathbb{R}$ for every fixed $a$ and $b$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point) — is continuous as a map $(\mathbb{R}^{2}, d_2) \to (\mathbb{R}, d_{\mathbb{R}})$ (Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions, Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form, $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it).

The witness. Define $g : \mathbb{R}^{2} \to \mathbb{R}$ by

$g(p) \;:=\; \frac{p_0\,p_1}{p_0^{2}+p_1^{2}} \ \text{ for } p \ne 0, \qquad g(0) := 0 ,$

writing $p = (p_0,p_1)$ for an element of $\mathbb{R}^{2}$, the set of functions $2 \to \mathbb{R}$ ($\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it). The quotient is defined for $p \ne 0$ because $p_0^{2}+p_1^{2} = \lVert p\rVert_2^{2} > 0$ there (The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).

Then $g$ is continuous in each variable separately at every point, and $g$ is not continuous at $0$.

This is the first function on $\mathbb{R}^{2}$ whose continuity this library studies, and its domain is $\mathbb{R}^{2}$ with the published metric $d_2$, not an informal plane.

• What the sequence sees. Along the line $p_1 = p_0$ the value of $g$ is constantly $1/\iota(2)$ off the origin, and points of that line come arbitrarily close to the origin; along either axis the value is constantly $0$. So the two partial functions through the origin cannot detect what a general approach does, and that is the whole phenomenon.

• Separate continuity is strictly weaker, and no repair is proposed here. What the refuted claim would need is a hypothesis controlling the two variables together — joint continuity is exactly such a hypothesis, and it is what Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions defines. Nothing here claims that any weaker hypothesis suffices.

• Nothing is claimed about $g$ away from the origin. The refutation needs only the behaviour of $g$ at $0$ together with the two partial functions, and that is all that is proved. In particular this item does not assert that $g$ is continuous at the points $p \ne 0$, true though that is; establishing it would need an algebra of continuous real-valued functions on a metric domain, which A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions provides for sums, scalar multiples and inner products but not for quotients.