Is there a function g continuous in the domain $(0,1)$ with range $R=[0,1]$.
Explain the answer.
This is a first semester calculus question, therefore I am curious about the depth someone has to reach to prove this one.
I believe that simple stating that for any function f that in domain $(0,1)$ $$\lim_{x\to z^-} f(x)= \lim_{x\to z^+} f(x) =f(z)$$
is true $\forall z\in (0,1)$ because the Range has no discontinuities is not enough.
Best Answer
To be honest the phrasing of the question makes very little sense. Let me try to answer what I think you are asking:
1) Yes there exists such function. Take for example $f(x)=\frac{1}{2}(1+\sin(2πx))$.
2) Continuity of $g$ doesn't guarantee that a function has that property. Indeed $g(x)=1$ doesn't have that property.