I have very big confusions on Continuous, differentiable and continuous isomorphism, homomorphism of a FUNCTION. I am seriously looking a single example, which can explain the following clearly.
Given an example of some single function which is
1) Continuous 2) Differentiable 3) Continuous Isomorphism 4) Continuous Homomorphism
and explain how that function satisfies each of the above.
Many many thanks! IFFl-edi
1) Roughly, a continous function is any function you can draw without lifting your pencil. Examples are $\sin,\cos,x,e^x$. A non-example is $$f(x) = \begin{cases} 1 & \text{ if } x \leq 0 \\ 0 & \text{ if } x < 0. \end{cases}$$
2) A differentiable function is any function without "kinks". Examples are all the above-mentioned functions. A non-example is the function $$f(x) = \mid x \mid = \begin{cases} x & \text{ if } x \leq 0 \\ -x & \text{ if } x < 0. \end{cases}$$
3) Formally, it is just a continous function that also has a continous inverse. Examples are $e^x$ (consideres as a function $\mathbb R \to (0,\infty)$ and $x \mapsto x$ (considered as a function $\mathbb R \to \mathbb R$). Non-examples are $\sin,\cos$ (these are not injective), depending on your choice of range and domain.
4) What is a homomorphism of topological spaces? Are you considering Lie groups?
ADDED Let $f:\mathbb R^{+} \to \mathbb R^+$ be defined by $f(x)=\frac 1x$.Then it is not hard to see that it is an homeomorphism. We can do this by checking that it has an inverse that is continous. But clearly $g(x) = \frac 1x$ from $\mathbb R^+ \to \mathbb R^+$ is an inverse. It remains to see that they both are continous: but this follows from the more general fact that if $h$ is cont's, then $1/h$ is also cont's.