Give an example of a continuous function which is defined on $(0,1]$ with range $(0,1)$.

I think it might be possible if there is a function defined on $(0,1]$ and oscillating between $(0,1)$, but how to write the function explicitly?


You can use $f(x)=\frac{(1-x)\sin(1/x)+1}{2}$.

You know that $|\sin(y)|\leq 1$, so, when we multiply it by $(1-x)$, for $x\in(0,1]$ we get a smaller number in absolute value. Therefore $|(1-x)\sin(1/x)|<1$. This implies that $0<((1-x)\sin(1/x)+1)/2<1$.

Now, in $(0,1)$ there are infinitely many points arbitrarily close to $0$ for which $\sin(1/x)=1$ and for likewise for which $\sin(1/x)=-1$.

Therefore, following a such a sequence of points we get either $((1-x)\sin(1/x)+1)/2\to (1\cdot 1+1)/2=1$, following a sequence of points in which $\sin(1/x)=1$ or $(1\cdot (-1)+1)/2=0$, following a sequence of points in which $\sin(1/x)=-1$. Therefore, the function attains values arbitrarily close to $0$ and to $1$. Since the function is continuous in $(0,1]$ all values in $(0,1)$ are attained.

The class of continuous functions is quite large. So, there is a lot of freedom to choose examples from. The whole idea above was to pick a function such that $f(1)$ is inside $(0,1)$, say $1/2$. And then such that towards $x=0$ it covers larger and larger portions of $(0,1)$.

You can achieve the same behavior by defining $f(1)=1/2$, $f(1/2n)=1-1/n$, $f(1/(2n+1))=1/n$, and joint these points with straight lines to fill up the graph of $f$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.