Suppose we have real function $$ \vert u'(t)\vert \leq ct^{-1} $$ where $c$ is a constant, and $t\in(0,\infty)$. Shall we have some properties of the function $u(t)$? More precisely, can we infer the range of $u(t)$? Thank you.
2026-03-26 01:28:47.1774488527
On the inequality $\vert u'(t)\vert \leq ct^{-1}$
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If we are talking real functions, we could have $u (t)=\log t $ (with range $\mathbb R $), or $u (t)=c $, with range a single point.