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I am trying to show if $f$ is continuous $[a,b]$ and its lower derivative $\underline{D}f$ is nonnegative on $(a,b)$, then $f$ is increasing on $[a,b]$. Here $\underline{D}f$ is defined as $$ \underline{D}f(x) = \lim\limits_{h \to 0} \inf\limits_{0 < |t| \leq h} \frac{f(x+t) - f(x)}{t} $$

[Hint in the textbook: first show this for a function $g$ for which $\underline{D}g$ is positive. Apply this to the function $g(x)=f(x)+\epsilon$x.]

In Calculus, we can show the global property by mean value theorem. I am wondering what theorem or definitions should I use here?

A similar question could be found here:

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Using the hint, for the function $g$ there is around each $x$ a radius $h_x$ such that the quotient is strictly positive. Now use the compactness of $[a,b]$ show that there is a finite set of $x_i$ such that the intervals $(x_i - h_{x_i}, x_i + h_{x_i})$ cover all of $[a,b]$. Since the intervals must overlap to cover $[a,b]$, you can show $g$ is increasing everywhere.

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  • $\begingroup$ where should I use the continuity? $\endgroup$ – yishen Oct 10 at 3:04
  • $\begingroup$ The rest of the conditions say nothing about the behavior of $g$ at $a$ and $b$ $\endgroup$ – Paul Sinclair Oct 10 at 16:38

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