A bounded function of two real variables whose every coordinate slice is real analytic is continuous

The false claim is the real-variable analogue of [L1], so refuting it shows that the several-variable holomorphic theorem is genuinely complex-analytic and not a formal separate-regularity statement.

The displayed formula defines a real-valued function on all of ${\mathbb{R}}^2$.

The function is bounded: for $(x,y)\mathrm{\ne }(0,0)$, the inequality $2\mathrm{\mid }xy\mathrm{\mid }\le x^2+y^2$ gives $\mathrm{\mid }f(x,y)\mathrm{\mid }\le 1\mathrm{/}2$, and at the origin $f(0,0)=0$, so $\mathrm{\mid }f(x,y)\mathrm{\mid }\le 1\mathrm{/}2$ everywhere.

The function is not continuous at $(0,0)$: if it were, then by [L3] applied with $\epsilon =1\mathrm{/}4$ there would be $\delta >0$ such that $\mathrm{\parallel }(x,y)\mathrm{\parallel }<\delta$ implies $\mathrm{\mid }f(x,y)\mathrm{\mid }<1\mathrm{/}4$; but for any real $t$ with $0<\mathrm{\mid }t\mathrm{\mid }<\delta \mathrm{/}\sqrt{2}$ one has $\mathrm{\parallel }(t,t)\mathrm{\parallel }=\sqrt{2}\mathrm{\mid }t\mathrm{\mid }<\delta$ and $f(t,t)=1\mathrm{/}2$, a contradiction.

Fix $b\mathrm{\ne }0$. Then the slice $x\mapsto f(x,b)=bx\mathrm{/}(x^2+b^2)$ is a quotient of polynomials, and the denominator never vanishes on $\mathbb{R}$, so [L2] makes this slice real analytic on all of $\mathbb{R}$. The same argument applies to the slice $y\mapsto f(a,y)$ when $a\mathrm{\ne }0$.

If $b=0$, then $f(x,0)=0$ for every real $x$, so the slice $x\mapsto f(x,0)$ is identically zero and therefore real analytic; by the symmetry $f(x,y)=f(y,x)$, the same is true of the slice $y\mapsto f(0,y)$. Thus every coordinate slice of $f$ is real analytic.

So this bounded function has real-analytic slices in each variable but is not jointly continuous, and the false claim fails. The published B-page item named cex-partial-derivatives-without-continuity uses the same witness for a different failure, but it is homed on an examples page and is therefore named here only in prose, without a wikilink or a dependency edge. The complex case is different precisely because [L4] turns a one-variable bound on a holomorphic slice into a uniform derivative bound, and no real-analytic hypothesis supplies such a bound.