I have been stuck on this Real Analysis problem for hours and am just totally clueless- I am sure it is some application of the Intermediate Value Theorem-
suppose $\ f: \mathbb{R}\rightarrow\mathbb{R} $ is continuous at every point. Prove that the equation
$ \ f(x) = c $ cannot have exactly two solutions for every value of $\ c. $
Would appreciate some help
Thanks
Best Answer
If every value of $f$ occurs exactly once, there is nothing to prove.
Otherwise, let $c$ be such that $f(a)=f(b)=c$ for $a<b$.
If $f$ is constant in $[a,b]$, then we are done, because $c$ is attained at least 3 times.
Otherwise, $f$ attains its maximum $M$ at an interior point $u$ in $[a,b]$. The value $M$ is then a local maximum. (This fails if $M=c$, but in this case take the minimum instead.)
If $f$ does not attain the value $M$ at another point in $\mathbb R$, then we are done.
Otherwise, let $v\ne u$ be such that $f(v)=M$ ($v$ is not necessarily in $[a,b]$).
For small enough $\epsilon>0$, the value $M-\varepsilon$ is then attained at least 3 times: twice near $u$ and at least once near $v$.