Final answer:
A function with a positive derivative almost everywhere typically suggests it is increasing, but one must check for points of discontinuity or where the derivative does not exist. Overall, if the function does not change from increasing to decreasing at these points, it can often be considered increasing.
Step-by-step explanation:
The question is whether a function with a positive derivative almost everywhere is an increasing function. If a function has a positive derivative at almost every point, it generally means that the tangent line to the function at those points has a positive slope, which suggests that the function is increasing at those points. However, if the function's derivative is not defined at some points or if there are discontinuities, the behavior of the function could be more complex.
To determine whether the function is truly increasing, we would consider any points where the derivative does not exist or any intervals where the function might not be continuous.
If there are only a finite number of such points and the function does not drastically change its behavior at those points (like changing from monotonically increasing to decreasing), the function can still be considered as increasing. This is because the overall trend of the function is to increase wherever the derivative is defined and positive. Nevertheless, a rigorous analysis would be required to confirm that the function is indeed overall increasing.
An example to illustrate this might be the function f(x) = x2sin(1/x) for x ≠0 and f(0) = 0. This function has a positive derivative almost everywhere, but it oscillates infinitely as x approaches zero. However, despite these oscillations, the overall trend of the function is increasing as x moves away from zero.