continuity of multivariable functions

• February 9th 2009, 12:22 PM
Greengoblin
continuity of multivariable functions
For proving continuity of single variable functions of the form $f:\mathbb{R}\to\mathbb{R}$ there is the epsilon-delta method, but for multivariable functions of the form $f:\mathbb{R}^n\to\mathbb{R}$ where $n\ge 2$, we need to show that $\lim_{(x_1,...,x_n)\to (a,...,n)}f(x_1,x_2,...,x_n)=f(a,b,...,n)$ from all possible directions including curves in $\mathbb{R}^n$ that pass through (a,b,...,n) so how do we go about proving continuity?
• February 9th 2009, 01:25 PM
Plato
Quote:

Originally Posted by Greengoblin
for multivariable functions of the form $f:\mathbb{R}^n\to\mathbb{R}$ where $n\ge 2$, we need to show that $\lim_{(x_1,...,x_n)\to (a,...,n)}f(x_1,x_2,...,x_n)=f(a,b,...,n)$ from all possible directions including curves in $\mathbb{R}^n$ that pass through (a,b,...,n) so how do we go about proving continuity?

In another post you said that you are studying basic point set topology.
But I will answer this question for metric spaces.
Basically we replace the open interval $\left( {a - \delta ,a + \delta } \right) = \left\{ {x:\left| {a - x} \right| < \delta } \right\}$ with open balls $B(p;\delta ) = \left\{ {x:d(x,a) < \delta } \right\}$.

In other words, there is a natural extension of the $\varepsilon ;\delta$ definition.

As we move into abstract topological spaces, we replace open balls with open sets.

I hope this very simplified outline gives you some guidance.
• February 10th 2009, 01:37 AM
Greengoblin
Quote:

Originally Posted by Plato
In another post you said that you are studying basic point set topology.
But I will answer this question for metric spaces.
Basically we replace the open interval $\left( {a - \delta ,a + \delta } \right) = \left\{ {x:\left| {a - x} \right| < \delta } \right\}$ with open balls $B(p;\delta ) = \left\{ {x:d(x,a) < \delta } \right\}$.

In other words, there is a natural extension of the $\varepsilon ;\delta$ definition.

As we move into abstract topological spaces, we replace open balls with open sets.

I hope this very simplified outline gives you some guidance.

Ah thanks, I have seen that in the topology notes I'm reading. I've only just started learning point-set topology and so far in my notes I've only read through the preliminaries. I actually didn't know it dealt with continuity of mulitvariable functions, but that gives me more motivation to learn it.