Math Challenge - April 2020

[Infrared]

Is there any function that can be expressed using only common standard functions and does not need to be written as a sum of infinite series that is continuous and differentiable in $(0,1)$ which has all rational numbers in its range and no other value as its zeros?

The range of any continuous map $f:(0,1)\to\mathbb{R}$ is connected, so if it contains $\mathbb{Q}$, then it must be all of $\mathbb{R}.$ There are continuous surjections like this, e.g. $f(x)=\tan(\pi(x-1/2)).$ What do you mean by "no other values as its zeros"?

Thanks for that example! It is nice to be able to prove something using just one simple theorem. But I am a bit confused about this function's relation to @wrobel's. @wrobel's function uses absolute value of $f(x)$ whereas yours does not, so your example appears to be of similar form but not exactly the same as the exponential of @wrobel's function. Am I missing something?

Okay, you're right, the exponential of wrobel's function is $e^{-x}|f(x)|.$ But since he is working on an interval where $f$ has no zeros, this is the same up to a possible minus sign. The point is that $f$ and $e^f$ have the same critical points, and negating doesn't change this.