91Ó°ÊÓ

A two-variable function f is known to be continuous everywhere.

The goal is to demonstrate why the function $\frac{f(x,y)}{x-y}$is continuous only if and only if $x≠y$

The continuity of functions asserts that if functions $f(x,y)andg(x,y)$are continuous in a certain interval, then the quotient function $\frac{f(x,y)}{g(x,y)}$is likewise continuous in the same interval, if and only if $g(x,y)≠0$.

For the function $\frac{f(x,y)}{x-y}$, we use this rule.

Everywhere, the function $f(x,y)$is said to be continuous.

A polynomial function is the denominator function $(x-y)$. As a result, it is also consistent throughout.

The sole remaining criterion is that the denominator does not equal $0$.

$$\begin{gathered} x-y≠0 \\ x≠y \end{gathered}$$

The goal is to show why for every real numbera, $\underset{(x,y)→(a,a)}{\lim }\frac{f(x,y)}{x-y}$does not exist.

The denominator of the above function becomes $0$at the point $(a,a)$. As a result, the function is indefinite. As a result, the limit doesn't exist at this point.