Exercise 1192Topology › Topology of ℝⁿL1Added 3 Nov 2026

Continuity versus iterated limits

#iterated limits#continuity#subtleties

Statement

Show that $f(x,y)=\dfrac{xy}{x^2+y^2}$ (with $f(0,0)=0$ ) has iterated limits $0$ along both axes but no limit at the origin.

Hint

Compute limits along $y=0$ and $x=0$ , then along $y=x$ .

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Iterated limits. $f(x,0)=0$ , $f(0,y)=0$ : so $\lim_{x\to0}\lim_{y\to0}f=\lim_{y\to0}\lim_{x\to0}f=0$ .

Diagonal. Along $y=x$ : $f(x,x)=\frac12$ , so the limit along this path is $1/2$ .

Conclusion. Iterated limits exist and coincide, but the two-dimensional limit does not: $f$ is not continuous at the origin.