For Continuous $f: [1,4] \to \mathbb R$, and given that $f(1)=f(4)$, Show there exists at least one $c \in [1,4]$ such that $f(c) = f(c+ 1.5)$.
(this is not homework but I am practicing some questions so a hint would also suffice)
My attempt:
Let $f(2.5)=k$. From IVT, for every $x \in [1,2.5]$ there is at least one $y \in [2.5, 4]$ such that, $f(x)=f(y)$. Since many such $y$ are possible let us restrict to $\inf y$.
Now, $y-x$ is continuous (because of continuity of f) and takes value from $d$ to $0$ where $d \geq 1.5$. So 1.5 is in this set.
Is this proof correct? Is there a simpler proof?
