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In the following video we go over the definitions of limit and continuity. The definitions are repeated in text form below the video. For functions of one variable, the (rough) definition of a limit was:
We made that precise by saying exactly what `close to' means. We used the letter $\epsilon$ for how close $f(x)$ has to get to $L$, and $\delta$ for how close $x$ is to $a$:
The rough and precise definitions of limits of functions of two (or more) variables work the same way:
The only difference is what we mean by `close to $(a,b)$'. We mean the distance in the plane: $$\\langle xa, yb \rangle\ = \sqrt{(xa)^2 + (yb)^2}.$$This is small whenever $x$ is close to $a$ and $y$ is close to $b$. Just one of them being close isn't good enough.
Just as with one variable, we say a function is continuous if it equals its limit:
Most of the rules for continuous functions carry over unchanged from single variable calculus. For instance,
