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.