Limit Continuity And Differentiability Ques 52

  1. Let $R$ be the set of real numbers and $f: R \rightarrow R$ be such that for all $x$ and $y$ in $R, f(x)-\left.f(y)\right|^2 \leq(x-y)^3$. Prove that $f(x)$ is a constant.
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Solution: Since,

$\Rightarrow \quad \frac{|f(y)-f(x)|^2}{(y-x)^2} \leq(x-y)$

$\Rightarrow \quad\left|\frac{f(y)-f(x)}{y-x}\right|^2 \leq x-y$

$\Rightarrow \quad \lim _{y \rightarrow x}\left|\frac{f(y)-f(x)}{y-x}\right|^2 \leq \lim _{y \rightarrow x}(x-y)$

$\Rightarrow \quad\left|f^{\prime}(x)\right|^2 \leq 0$

which is only possible, if $\left|f^{\prime}(x)\right|=0$

$\therefore \quad f^{\prime}(x)=0$

or $f^{\prime}(x)=$ Constant



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