Let f be a differentiable function from R to R such that |f(x)-f(y)|2|x-y|^3/2, for all x,…

Question

Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$ such that $|f(x)-f(y)|\le2|x-y|^{3/2}$, for all $x,y\in\mathbb{R}$. If $f(0)=1$ then $\displaystyle\int_0^1f^2(x)\,dx$ is equal to :

Accepted Answer

From $|f(x)-f(y)|\le2|x-y|^{3/2}$:
$$\left|\frac{f(x)-f(y)}{x-y}\right|\le2\sqrt{|x-y|}$$

Taking $x\to y$, the right side $\to0$, forcing $f'(y)=0$ for all $y$. So $f$ is constant, and since $f(0)=1$, $f(x)=1$ for all $x$.

$$\int_0^1f^2(x)dx=\int_0^1 1\,dx=1$$

Solutions · 1