Assume $f$ is a smooth (e.g. $C^1$) real-valued function defined on a domain $\Delta\subset\mathbb{R}^n$. Let $t\in[a,b]\to\gamma(t)$ be a smooth curve in $\Delta$ with endpoints $x=\gamma(a)$ and $y=\gamma(b)$. Then, $$f(y)-f(x)=\int_a^b\nabla f(\gamma(t))\cdot\gamma'(t)dt,$$ from which follows, together with Cauchy-Schwarz inequality, that $$|f(y)-f(x)|\leq\sup_{s\in\gamma}\|\nabla f(s)\|_2\cdot\text{length}(\gamma). $$ Question : Does this inequality still hold true if we only assume the curve $\gamma$ to be rectifiable ?
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