Question

Is the function f:R^2-{(0,0}→R^2 defined by
f(x,y)=(x/(x^2+y^2 ),-y/(x^2+y^2 ))
continuous on R^2-{(0,0}?

Answer 1

Answer on Question #42158 – Math -Real Analysis

Is the function $f\colon \mathbb{R}^2\backslash \{0,0\} \to \mathbb{R}^2$ defined by $f(x,y) = \left(\frac{x}{x^2 + y^2}, - \frac{y}{x^2 + y^2}\right)$ continuous on $\mathbb{R}^2\backslash \{0,0\}$?

Solution.

Theorem. Let $f \colon A \subset \mathbb{R}^n \to \mathbb{R}^m$ be given by

$$f (x) = \left(f _ {1} (x); \dots ; f _ {m} (x)\right).$$

Then $f$ is continuous at a $a \in A$ if and only if $f_{i}$ is continuous at $a$ for $i = 1,2,\ldots ,m$.

So it is enough to show that functions $f_{1}(x,y) = \frac{x}{x^{2} + y^{2}}$ and $f_{2}(x,y) = -\frac{y}{x^{2} + y^{2}}$ are continuous on $\mathbb{R}^2\backslash \{0,0\}$. They are continuous as a fraction of the two continuous functions.

Answer: Yes, It is.

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