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.

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# Continuity of multivariable function

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