Interval partition is a crucial concept in the field of integrals in real analysis. A partition of a closed and bounded interval is, by definition, a strictly increasing sequence of numbers starting from the initial point of the interval and reaching to the final point of the interval. Definition of interval partition An interval partition…

# Month: May 2023

## Uniform continuity

The main purpose of this post is to introduce the concept of uniform continuity. A function satisfies the uniform continuity if the values of the function are close to each other for all elements sufficiently close in its domain. More precisely, a function $$f : D \subseteq \mathbb{R} \rightarrow \mathbb{R}$$ satisfies the uniform continuity if…

## L’Hôpital’s rule

In calculus, the L’Hôpital’s rule states that the limit of a quotient of functions is equal to the limit of its derivatives, under certain conditions. A statement and a proof of the L’Hôpital’s rule for 0/0 form Result (L’Hôpital’s rule). Let \(f\) and \(g\) be real-valued differentiable functions over an open interval \(I\) except possibly…

## First derivative test

The first derivative test gives a criterion to identify points in the domain of a function at which it has extreme values. In this post, we state and prove the first derivative test and we use this test to find extreme values of some functions. A statement and a proof of the first derivative test…

## Mean value theorem

The mean value theorem is an essential result in calculus and real analysis that expresses a relationship between the derivative of a function and its average rate of change. More precisely, the mean value theorem states that if a function is continuous over a closed interval \(I = [a,b]\) and differentiable on its interior \((a,b)\),…

## Rolle’s theorem

The Rolle’s theorem states that the derivative of any real-valued differentiable function attaining equal values at two distinct points will vanish at some point between them. More precisely, let \(f\) be a real-valued function continuous on \([a,b]\) and differentiable on \((a,b)\) with \(f(a) = f(b)\). Then, the Rolle’s theorem states that there is a point…

## First derivative theorem

One of the most basic and essential results in differential calculus known as the “first derivative theorem” states that the derivative of a function at an extremum in an open interval inside the domain of the function vanishes. Note that in the post on the extreme value theorem, we showed that a continuous real-valued function…