True or False There are infinitely many continuous functions f for which $\int_0^1f(x)(1-f(x))dx=\frac{1}{4}$ https://ift.tt/eA8V8J

I was going through CMI 2019 paper, I stumbled upon the statement:

For $f:R\rightarrow R$, There are infinitely many continuous functions f for which $\int_0^1f(x)(1-f(x))dx=\frac{1}{4}$

My approach is, we can write $f(x)(1-f(x))=\frac{1}{4}-(f(x)-\frac{1}{2})^2$

Now, we can write $\int_0^1f(x)(1-f(x))dx=\frac{1}{4}-\int_0^1(f(x)-\frac{1}{2})^2dx$

$\Rightarrow \int_0^1(f(x)-\frac{1}{2})^2dx=0 \Rightarrow f(x)=\frac{1}{2}$

So, there must be only one function satisfying the condition. However, the official answer says the statement is true. Am I missing something?

from Hot Weekly Questions - Mathematics Stack Exchange
Mathsmerizing