# Continuous function with nonnegative lower derivative is increasing.

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:

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.

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