In general I have a problem to recognise if a function is continuous or not. I simply don't know where I should start to actually see it.
Here there is an example of my problem that I found in a proof in a book.
Let $\Delta$ be the Cantor space and define the set $C$ as
$$ C:= \{ \sum_{n=1}^\infty\frac{a_n}{3^n} : a_n=0 \text{ or } a_n=2 \}. $$
Proposition: The Cantor set $\Delta$ is homeomorphic to $C$.
Proof: Define $$\phi (a_1, a_2, \dots ) = \sum_{n=1}^\infty \frac{2a_n}{3^n}.$$ Then $\phi$ is continous, one-to-one and surjective. Hence, it is a homeomorphism. $\square$
Now, why exactly $\phi$ is continous?
I simply don't know where to start when I find statements like this one.
Any feedback or help is welcome.
Thank you for your time.
A function $\phi$ is continuous if for every $\epsilon>0$ there exists a $\delta>0$ so that if $\Vert x-y\Vert<\delta$ then $\Vert \phi(x)-\phi(y)\Vert<\epsilon$.
Suppose that $\Vert x-y\Vert<\delta$. Then $|x_{i}-y_{i}|<\delta$ for each $i$. Hence \begin{align*} |\phi(x)-\phi(y)| &= \left|\sum_{n=1}^{\infty} \frac{2x_{i}}{3^{n}} - \sum_{n=1}^{\infty} \frac{2y_{i}}{3^{n}}\right|\\ &=\left|\sum_{n=1}^{\infty}\frac{2(x_i-y_i)}{3^{n}}\right| \\ &\leq \sum_{n=1}^{\infty}\frac{2|x_{i}-y_{i}|}{3^{n}} \\ &< \sum_{n=1}^{\infty}\frac{2\delta}{3^{n}} \\ &= c\delta \\ &< \epsilon\end{align*} If $\delta < \epsilon/c$.