Quiz for Friday, January 20 (PDF)

This quiz was about level sets and continuity. The notion of continuity for calc 3 is really the same as it was way back in first semester calculus. Namely, you have three conditions for a function $f(x,y)$ to be continuous at a point $c$:

- The function must be defined at $c$. That is, if $c=(x_1,y_1)$, then you must be able to find $f(x_1,y_2)$.
- The limit of the function at $c$ must exist. That is, you should be able to find $\displaystyle\lim_{(x,y)\to(x_1,y_1)}f(x,y)$. Of course, calculating limits in multiple dimensions can be a hard thing to do. (Generally, the examples you’ll see in this course won’t be too paradoxical.)
- The function’s value and the value of the limit must agree at $c$. That is,

\[f(x_1,y_1)=\lim_{(x,y)\to(x_1,y_1)}f(x,y).\]

The book glosses over a bunch here by only requiring number 3 in the above list. Why? Well, technically speaking, by saying that the value of the function agrees with the value of the limit, you’re actually implicitly saying that the function is defined and the limit exists. (If something is to be equal to something else, it has to exist before equality even makes sense.) So, in the text, you’ll really just see number 3 above as the definition in multiple variables.

Of course, I used two variables here for the definition of continuity. You can extend this to as many variables as you need.

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