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Show that $f(x, y, z)=x+y-z$ is continuous at every point $\left(x_{0}, y_{0}, z_{0}\right)$

$\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}}<\delta \Rightarrow$ $\left|f(x, y, z)-f\left(x_{0}, y_{0}, z_{0}\right)\right|<\epsilon$

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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Lectures

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Show that $f(x, y, z)=x+y-…

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evaluate whether this function f of X y Z equals X plus y minus C is continuous at every point because it's a polynomial, then it will have to be continuous at every point. So therefore, yes, this is true and we could look at it a couple of different ways. We could look at it through the lens of. There's only two times that functions are not continuous when we divide by zero or if we take the even route. So some even power like two or four Q group or the fourth root of a negative number then, of course, either of these disrupts the continuity. Um, but neither of these is an issue here. And furthermore, we could look at just the case where we want to verify that the limit as ex why and see all approach. That's not why not Lindsay Not well, the US And just to verify that that limit of F um doesn't end up Equalling um f of x, not Why not see? Not as well, and all of those points will end up equaling their values there as well. So yes, this function is continuous. And these three variables these being polynomial, is there's nothing to disrupt the continuity

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