Continuity for Multivariable Functions

[brydustin]

Just curious how to define continuity for mult dim. functions. I know that the topological and set theorectical definitions work in a very abstract setting; but I just don't know how to prove (for example) that f(x,y) = x + y or f(t,z) = t*z is continuous, other than saying something like: Well because f(t)=t is continuous and ..., therefore the composition of... and hence it is continuous. I was wondering if there is a generalization for delta-epsilon defition that would cover any (actual) function (as opposed to some abstract space,etc): R to R and R^n to R^m (where m,n are any positive integers and may or may not be equal). Can we possibly express epsilon-delta as a vector? A stretch maybe? Thanks for any help,... I'm teaching myself topology and more analysis right now! got to love it!

 

[snipez90]

What do you mean an (actual) function as opposed to some abstract space? This doesn't seem to make much sense because we obviously need the point-set before we can define the function. In a metric space, we can define continuity in terms of the metric, and if we have a function f from R^n to R^m, we're just working with the usual topology. Thus continuity of f at a point a in R^n is just the same old "for every epsilon > 0 there exists a delta > 0 such that d(x, a) < delta implies d(f(x), f(a)) < epsilon" where x is in R^n and d is the standard euclidean distance.

 

[mathwonk]

what are you reading? Isn't this explained in every book on several variable calculus? i like courant, volume 2, and wendell fleming's book. basically you define the length of a vector |(a,b,c)| = sqrt(a^2+b^2+c^2), and do all the epsilon delta in those terms, word for word the same as in one variable.

It's differentiability where you have to do something new, because you can't divide vectors.|x+y| ≤ |x| + |y|, implies continuity of addition.

 

[LCKurtz]

Just curious how to define continuity for mult dim. functions. I know that the topological and set theorectical definitions work in a very abstract setting; but I just don't know how to prove (for example) that f(x,y) = x + y or f(t,z) = t*z is continuous, other than saying something like: Well because f(t)=t is continuous and ..., therefore the composition of... and hence it is continuous. I was wondering if there is a generalization for delta-epsilon defition that would cover any (actual) function (as opposed to some abstract space,etc): R to R and R^n to R^m (where m,n are any positive integers and may or may not be equal). Can we possibly express epsilon-delta as a vector? A stretch maybe? Thanks for any help,... I'm teaching myself topology and more analysis right now! got to love it!

Of course, even with functions of one variable, nobody would use the epsilon-delta kind of argument for an "actual" function something like

f(x) = \frac{e^{\sin(x)}\sqrt{1+x^3}}{3x^2+7}

You prove continuity for the basic functions and use the theorems about products, quotients, compositions etc. The same thing is true for functions of 2 variables.

 

[slider142]

but I just don't know how to prove (for example) that f(x,y) = x + y or f(t,z) = t*z is continuous, other than saying something like: Well because f(t)=t is continuous and ..., therefore the composition of...

Note that continuity along lines parallel to the z and t axes does not imply continuity of the function; consider
<br /> f(z,t) = \left\{\begin{array}{ll}<br /> \frac{|t|e^{-\frac{|t|}{z^2}}}{z^2} &amp; z\neq 0 \\<br /> 0 &amp; z=0 \end{array}<br />
in the neighborhood of (0,0). Although the limits along the axes both exist and are equal to the value of the function at (0,0), \lim_{(z,t)\rightarrow (0,0)} f(z,t) does not exist.

 

[brydustin]

Actually the book I'm using is Real Analysis by Norman B Haaser and Joseph A Sullivan. It explains the concept of continuity on metric spaces but doesn't given any examples other than those of one variable however wants the reader to do examples of more than one variable as practice problems.

So I still don't have a clue how to prove that x+y is continuous. Any more help?

what are you reading? Isn't this explained in every book on several variable calculus? i like courant, volume 2, and wendell fleming's book. basically you define the length of a vector |(a,b,c)| = sqrt(a^2+b^2+c^2), and do all the epsilon delta in those terms, word for word the same as in one variable.

It's differentiability where you have to do something new, because you can't divide vectors.


|x+y| ≤ |x| + |y|, implies continuity of addition.