Exercise 1602Calculus › Multivariable calculus › ContinuityL1Added 3 Nov 2026

A function that is not continuous despite radial limits

#continuity#radial limit#counterexample

Statement

Show that $f(x,y)=\dfrac{x^2y}{x^4+y^2}$ (with $f(0,0)=0$ ) is not continuous at $(0,0)$ , even though $f(x,ax)\to0$ for every $a$ .

Hint

Test the path $y=x^2$ .

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Radial limits. $f(x,ax)=\frac{ax^3}{x^4+a^2x^2}=\frac{ax}{x^2+a^2}\to0$ for fixed $a$ .

Parabolic path. $f(x,x^2)=\frac{x^4}{2x^4}=\frac12\not\to0$ .

Conclusion. $f$ is not continuous at the origin despite radial limits being zero.