Q 56. - Calculus

Question

Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous.  

$$\begin{gathered} f(x,y)=\left\{e^{-1/x^2+y^2} if (x.y)\ne (0.0)\right\} \\ \left\{0 if \left(x,y\right)=\left(0,0\right)\right\} \end{gathered}$$

Step-by-Step Solution

Answer

The given function is continuous everywhere.

Step 1. Given information.

We have given expression: $$\begin{gathered} f(x,y)=\left\{e^{-1/x^2+y^2} if (x.y)\ne (0.0)\right\} \\ \left\{0 if \left(x,y\right)=\left(0,0\right)\right\} \end{gathered}$$ 

Step 2: To find domain of the function.

The given function is a piece wise function, with a break point at $\left(x,y\right)=\left(0,0\right)$.

The given function can be write as

$$\begin{gathered} f(x,y)=\left\{\frac{1}{e^{1/x^2+y^2}} if (x.y)\ne (0.0)\right\} \\ \left\{0 if \left(x,y\right)=\left(0,0\right)\right\} \end{gathered}$$

An exponential function as well as its reciprocal is defined for all the values at which the exponent is defined. 

The exponent itself is a rational expression. 

The rational expression means that the denominator cannot be equal to 0. 

That is $x^2+y^2\ne 0$

The left side of this inequality is sum of two squares. 

Hence, it is always positive. 

The only case where it is not so, when x=y=0, but that is not the case with the given function. Hence, the function is defined for all the real values. 

Thus, the domain of the function is given as, 

$f^2 or \left\{\left(x+y\right):x.y\in f\right\}$

Step 3: To find continuous of the function.

Substitute the value of $x=r \cos \theta$ and $y=r \sin \theta$

$$\begin{gathered} \underset{\left(x,y\right)\to \left(0,0\right)}{\lim }\frac{1}{e^{1/x^2+y^2}}=\underset{r\to 0}{\lim }\frac{1}{e^{{\displaystyle \frac{1}{{(r\cos \theta )}^2+{(r\sin \theta )}^2}}}} \\ =\underset{r\to 0}{\lim }\frac{1}{e^{{\displaystyle \frac{1}{{r^2(\cos \theta }^2+{\sin \theta )}^2}}}} \\ =\underset{r\to 0}{\lim }\frac{1}{e^{{\displaystyle \frac{1}{r^2}}}} \\ =0 \end{gathered}$$

Hence, there is no point of discontinuity for the given number.