# Increasing and differentiable implies nonnegative derivative that is not identically zero on any interval

## Contents

## Statement

### On an open interval

Suppose is a function on an open interval that may be infinite in one or both directions (i..e, is of the form , , , or ). Suppose the derivative of exists everywhere on . Suppose further that is an increasing function on , i.e.:

Then, for all . Further, there is no sub-interval of such that for all in the sub-interval.

### On a general interval

Suppose is a function on an interval that may be infinite in one or both directions and may be open or closed at either end. Suppose is a continuous function on all of and that the derivative of exists everywhere on the interior of . Further, suppose is an increasing function on , i.e.:

Then, for all in the interior of . Further, there is no sub-interval of such that for all in the sub-interval.

## Related facts

### Converse

### Other similar facts

- Positive derivative implies increasing
- Zero derivative implies locally constant
- Negative derivative implies decreasing

## Facts used

- Local maximum from the left implies left hand derivative is nonnegative if it exists
- Local minimum from the right implies right hand derivative is nonnegative if it exists
- Zero derivative implies locally constant

## Proof

If is increasing on , then every point in the interior of is a point of local maximum from the left and local minimum from the right. Thus, by Facts (1) and (2), both the left hand derivative and the right hand derivative of , if they exist, are nonnegative at any point in the interior of . In particular, if the derivative itself exists at a point in the interior of , then it must be nonnegative at that point.

It remains to show that the derivative is not zero on any sub-interval of . For this, note that by Fact (3), a derivative of zero forces the function to be constant on that sub-interval. This, however, contradicts the definition of an increasing function. We thus have the desired contradiction and we are done.