Hi, I'm having trouble with the following question. I would like some help with it.(adsbygoogle = window.adsbygoogle || []).push({});

Q. A function [itex]f:A \subset R^n \to R^m[/itex] is continuous if and only if its component functions [itex]f_1 ,...,f_m :A \to R[/itex] are continuous.

Firstly, is there a difference between [itex]C \subset D[/itex] and [itex]C \subseteq D[/itex]? Anyway in this question I need to do both directions.

The definition I have of continuity is:

[itex]f:A \subset R^n \to R^m[/itex] is continuous at [itex]\mathop {x_0 }\limits^ \to \in A[/itex] if

[tex]\mathop {\lim }\limits_{\mathop x\limits^ \to \to \mathop {x_0 }\limits^ \to } f\left( {\mathop x\limits^ \to } \right) = f\left( {\mathop {x_0 }\limits^ \to } \right)[/tex]

Alternatively: given [itex]\varepsilon > 0[/itex], there exists [itex]\delta > 0[/itex] such that

[tex]\left\| {f\left( {\mathop x\limits^ \to } \right) - f\left( {\mathop {x_0 }\limits^ \to } \right)} \right\| < \varepsilon [/tex]

[tex]\forall x \in A[/tex] satisfying [itex]\left\| {\mathop x\limits^ \to - \mathop {x_0 }\limits^ \to } \right\| < \delta [/itex].

(yes, it does say x is an element of A where x is just the normal scalar variable - I would've thought that x would be a vector in R^n in this case)

I'm not really sure where I should start. If I was to begin with f being continuous then I could write an equation saying that the limit as an arbitrary n-vector (call it (x_1,..,x_n)) approaches a fixed n-vector (call it (b_1,...,b_n)), is equal to f applied to the fixed n-vector. To show one side of the implication I would need to deduce that

[tex]\mathop {\lim }\limits_{x_i \to b_i } f\left( {x_i } \right) = f\left( {b_i } \right)[/tex] for each i = 1,...,n.

It just seems so immediate that continuity of f automatically leads to continuity of its components and vice versa. Is this just a matter of writing down a few equations or is it more complicated? Any help would be good thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Continuity of multivariable function

**Physics Forums | Science Articles, Homework Help, Discussion**