- #151

Infrared

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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"?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 hasallrational numbers in its range and no other value as its zeros?

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.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?