I previously asked this question and was told that an answer is certainly possible but I am still looking for an example.
The question was for cases when the intermediate value theorem is true and a continuous function takes all values between f(a) and f(b) at least ones, is there ever a case when a particular value is attained infinitely many times? I know the trivial cases like f(x)=5 but something non-trivial, where $f(a)\neq f(b)$. I think the conclusion was that uncountably many times isn't possible but countably infinitely many times is certainly possible. Thanks!