Multivariable Continuity problem 1 by Dhaval Furia

Calculus Level 2

A function $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is defined as follows.

$f(x,y) = \frac {sin(x^{2} + y^{2})}{x^{2} + y^{2}}$ if $(x,y) \neq (0,0)$ &

$f(x,y) = a$ if $(x,y) = (0,0)$

Then which of the options is true ?

f is continuous at (0,0) if a = 1 f is continuous at (0,0) if a = 0 f is continuous at (0,0) if a = -1 f is not continuous at (0,0) for any value of a

1 solution

Stephen Brown
Nov 28, 2017

Clearly f is continuous everywhere other than (0,0). For f to be continuous at (0,0), we must have

$a=\lim_{(x,y) \rightarrow (0,0)}f(x,y)$

Evaluating multivariate limits can be tricky; in a single variable function, it suffices to show that the limits from both the left and the right exist and are equal. When dealing with two variables, we must somehow show that the limit exists from ALL directions and is always a single value. In this particular problem, we can make a simple substitution that will reduce the function to one variable: polar coordinates.

$x = r\cos(\theta), y=r\sin(\theta) \Rightarrow f(r,\theta) = f(r) = \frac{\sin(r^2)}{r^2}$

Now we can evaluate $\lim_{r \rightarrow 0^+}f(r)$ , which is a well-known limit that can be evaluated using L'Hospital's rule. The limit is equal to 1. Thus,

$\lim_{(x,y) \rightarrow (0,0)}f(x,y) = \lim_{r \rightarrow 0^+}f(r) = 1 \Rightarrow \boxed{a=1}$

Multivariable Continuity problem 1 by Dhaval Furia

1 solution

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