existence, continuity and limit; cardinality

ma_go

1.let's consider the equation $\ln x = (\cot x)^\alpha$ with $\alpha \in (0, +\infty)$ and $x \in (0,{\pi \over 2})$ .
a. show that for any $\alpha$ there exist one and only one solution, and let us denote it with $s(\alpha)$ .
b. show that $s$ is continuous.
c. find $\mathop {\lim }\limits_{\alpha \to 0^+ }x_\alpha$ .

2. Find the cardinality of the set of the real continuous functions.

**Posted:** Sun Feb 13, 2005 8:13 am

liyi

(2)

All constant functions are continuous. Hence the cardinality of continuous functions is $\geq c$

For continuous function $f$ , define
$g(f) = \{ (x,y)\in\mathbb{Q}\times\mathbb{Q} : y\leq f(x)\}$
It can be shown that $g$ is an injection from the set of continuous functions to $\mathcal{P}(\mathbb{Q}\times\mathbb{Q})$ . The cardinality of the latter set is $c$ . Hence the cardinality of continuous functions is $\leq c$ .

**Posted:** Sun Feb 13, 2005 6:33 pm

ma_go

ok, one has gone :)
what about the other one?

**Posted:** Mon Feb 14, 2005 12:31 pm