Mathematics - Continuity - Differentiability Question with Solution | TestHub

For any $x\in \mathbb{R}$, consider the derivative $f^{\prime }(x)={\lim }_{y\to x}\frac{f(y)-f(x)}{y-x}$. From the given condition, $0\le \left|\frac{f(y)-f(x)}{y-x}\right|\le M|y-x|$. Taking the limit as $y\to x$, we get $0\le |f^{\prime }(x)|\le {\lim }_{y\to x}M|y-x|=0$. This implies $f^{\prime }(x)=0$ for all $x\in \mathbb{R}$. A function whose derivative is zero everywhere on $\mathbb{R}$ must be a constant function.