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My instructor said that f(x,y) is continuous at (a, b)

if the limit exists for

$$\lim\limits_{(x,y) \rightarrow (a, b)} f(x,y)$$ and the limit is equal to f(a, b)

So can I simplify this statement to: If I plug in (a,b) into f(x,y) and I get a number, then the limit exists and there is continuity at that point?

And if that is true, and lets say I get a case where I plug (a,b) into f(x,y) and I get an indeterminant (or something that is not a number) which means there is no continuity

Then I can try to prove it DNE by using the technique about going along paths or I can try to prove it does exist with other various techniques.

What happens in other cases like if I plug (a,b) into f(x,y) and get 1/0 or something, Im not sure what other cases there are..

Is my thought process correct? What am I missing?

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  • $\begingroup$ No, of course it's not true. If it were, then the definition of continuity would have a bunch of "just plug in" statements. $\endgroup$ – zhw. Oct 27 '17 at 23:06
  • $\begingroup$ what part are u referring to $\endgroup$ – mathguy Oct 27 '17 at 23:21
  • $\begingroup$ The parts where you say "plug in" $\endgroup$ – zhw. Oct 27 '17 at 23:45

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