Proving Continuity in Metric Spaces | Sequential Characterisation of Continuity

[Ted123]

The sequential characterisation of continuity says that f is continuous at x_0 if and only if for every sequence (x_n)_{n\in\mathbb{N}} in X, f(x_n)\to f(x_0) as x_n \to x_0. f is continuous on X if this is the case for all x_0 \in X.

I think I've done all the parts of this question up to the last 2 parts.

For part (b) is this right:

Suppose (x_n)_{n\in\mathbb{N}} is a sequence in X with x_n \to x\in X. Then for all x\in X: f(x_n) = (f_1(x_n) , f_2(x_n) , ... , f_N (x_n)) \to (f_1(x) , f_2(x) , ... , f_N (x) ) = f(x) since all the f_i are continuous.

(This is also using a theorem which says that if (x^{(n)})_{n\in\mathbb{N}} is a sequence of vectors in \mathbb{R}^N then x^{(n)} \to x\in\mathbb{R}^N in the Euclidean metric \iff x_j^{(n)} \to x_j for each 1\leqslant j \leqslant N in the standard metric on \mathbb{R}.)

How would you show in the last 2 parts that F and H are continuous?

 

[Ted123]

EDIT: I see that F = \phi \circ f and we've already shown that the composition of 2 continuous functions is continuous.

What function can I compose F with to turn the product in H into a sum?

 

[spamiam]

Well, to turn a sum into a product we could use $e^x$ since $e^{x+y} = e^x e^y$. So what might you use to do the opposite?

 

[Ted123]

Well, to turn a sum into a product we could use $e^x$ since $e^{x+y} = e^x e^y$. So what might you use to do the opposite?

Log!

 

[spamiam]

Log!

Right! And now this explains why the functions must be strictly positive for the last part.

 

[Ted123]

Right! And now this explains why the functions must be strictly positive for the last part.

If h : (X,d_X) \to \mathbb{R}^N is defined by h(x) = (h_1(x) , h_2(x) , ... , h_N(x) ) then h is continuous by part (b).

\displaystyle \log \circ H (x) = \log(H(x)) = \log \left( \prod_{j=1}^{N} h_j(x)^{a_j} \right)= \sum_{j=1}^N \log ( h_j(x)^{a_j} )= \sum_{j=1}^N a_j \log(h_j (x)) = \phi \circ (\log \circ h (x))

\phi , \log , h are all continuous so their composition is continuous, so \log \circ H is continuous.

We know if f and g are 2 continuous functions then g \circ f is continuous but g \circ f continuous \not\Rightarrow f, g are continuous so how do I frame the argument to show that H is continuous?

 

[Deveno]

If h : (X,d_X) \to \mathbb{R}^N is defined by h(x) = (h_1(x) , h_2(x) , ... , h_N(x) ) then h is continuous by part (b).

\displaystyle \log \circ H (x) = \log(H(x)) = \log \left( \prod_{j=1}^{N} h_j(x)^{a_j} \right)= \sum_{j=1}^N \log ( h_j(x)^{a_j} )= \sum_{j=1}^N a_j \log(h_j (x)) = \phi \circ (\log \circ h (x))

\phi , \log , h are all continuous so their composition is continuous, so \log \circ H is continuous.

We know if f and g are 2 continuous functions then g \circ f is continuous but g \circ f continuous \not\Rightarrow f, g are continuous so how do I frame the argument to show that H is continuous?

what continuous function can you compose with log \circ H to recover H?

 

[Ted123]

\exp \circ ( \log \circ (H(x) ) = H(x) = \exp \circ (\phi \circ (\log \circ h(x)) )

\exp , \phi , \log , h are all continuous so their compositions are all continuous, so H is continuous.

 

[Deveno]

i'll buy that.