I have two continuous functions $f(x)$ and $g(x)$ in a close and limited interval $[a,b]$. There are a series of questions that can be true or false. I just wanted to clarify if there are specific theorems or if there are only false examples to check if the assertions are true or false.
- $\sqrt{f(x)}$ for my opinion being the domain $f(x)\geq 0$ for all $x\in[a,b]$, surely it is continue function. If I not put $f(x)\geq 0$ it is false.
- $|f(x)|$ for my opinion is a continue function: an example it is the function $|x|$;
- $|f(x)-g(x)|$, if $f$ and $g$ are continuous function also $\psi=f-g \ $ is a continue function. Hence $|f(x)-g(x)|$ is continue function.
- $\sqrt[2n]{g(x)}$ with $n\in \mathbb{N}, n>0$; if I not specify the domain of $g(x)$ then it is false.
- $\sqrt[2n+1]{g(x)}$ with $n\in \mathbb{N}, n>0$ it is always continue function.
If $h=h(x)$ is continue $\forall x\in \Bbb R$, hence
- $(h\circ h)(x)=h(h(x))$ for my opinion it is true (composition of continuous functions are always continuous);
- $\varphi=\arcsin(h(x))$ is it continue on $[-1,1]$? The domain of $\varphi$ is $-1\leq h(x)\leq 1$ and for humble opinion it is false because all depends of the solution $-1\leq h(x)\leq 1$. For example $\varphi=\arcsin\left(x+1\right)$ is not continuous in $[-1,1]$.
- $\ln(f(x))$ is it continuous $\forall x\in \Bbb R$? False, because $f(x)>0$ and may not be true for every $x \in \Bbb R$.
- $\cos(f(x))$ is it continuous $\forall x\in \Bbb R$? True.