In real analysis, the Riemann sum of functions is a specific kind of approximation calculated from a finite sum. The Riemann sum of functions is used for the definition of Riemann integrability which has applications in calculating “the area under a curve”, “the length of a curve”, and “the solid of revolution”. Definition of the…

## Interval partition

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…

## 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…

## Differentiation rules

It is possible to calculate the derivatives of single variable functions using differentiation rules without taking limits each and every time we compute the derivatives. In this post, we introduce several rules of differentiation. However, we prove only a few of them and for the proof of the other rules, the reader should refer to…

## One-sided derivatives

One-sided derivatives are defined by one-sided limits. In this post, we will introduce the right-hand derivative and the left-hand derivative of a function at a given point. One-sided limits Let \(f\) be a real-valued function. The reader should be able to easily prove the following result: Result. Let \(f\) be a real-valued function, \(I\) an…