Exercise 869Calculus › Multivariable calculus › ContinuityHigh schoolAdded 3 Nov 2026

Continuous extension

#continuous extension#removable singularity

Statement

Can $f(x,y)=\dfrac{\sin(xy)}{xy}$ be extended by continuity at $(0,0)$ ?

Hint

Use $\sin t/t\to1$ .

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Limit. Along $(x,y)\to(0,0)$ with $xy\neq0$ , $\frac{\sin(xy)}{xy}\to1$ .

Extension. Define $f(0,0)=1$ ; $f$ becomes continuous.

Conclusion. Yes, extension by $f(0,0)=1$ exists.