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I have a special problem here related to proving that a function is continuous.

We have:

$$f(x,y)=e^{\frac{x-1}{y}+\frac{y}{x-1}}\sqrt{1+\frac{x-y}{x+y}}$$

which shows that the function does not exist at the line $y$ and not at the line $x=1$, nor on the plane $x=-y$.

How do I prove that this is nevertheless continuous on its domain?

The two factors are both containing forbidden regions, so how does one prove continuity for this hard nut?

Thanks!

Dole
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    You mean to say "the line $y=0$" and "the line $x=-y$." – saulspatz Mar 30 '21 at 13:39
  • Correct, yes the line y=0 –  Mar 30 '21 at 13:40
  • The product of continuous function is continuous. You also have to exclude points where the argument of the square root would be negative. – saulspatz Mar 30 '21 at 13:41
  • The negative arguments, indeed. Sorry I forgot about that. However, writing that the function is continuous on its domain after being defined as such, as there are no other points of inexistence, is it sufficient as a proof? –  Mar 30 '21 at 13:41
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    Yes, certainly. The exponential and square root are continuous wherever they're defined, so where both are defined, $f$ is the product of continuous functions, hence continuous. – saulspatz Mar 30 '21 at 13:44
  • Ok, that was clear! Thanks! –  Mar 30 '21 at 13:45
  • @saulpatz defining the points that yield negative argument in the root is quite tricky for this case. Can one sufficiently say that it is not existing where $-\frac{x-y}{x+y}>1$ ? –  Mar 30 '21 at 13:47
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    It isn't really very tricky. Consider the cases $x+y>0$ and $x+y<0$ separately. As to your question, it depends on what your teacher thinks, so I can't answer. – saulspatz Mar 30 '21 at 13:50

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