The concept of a limit is fundamental to calculus. In this unit we will introduce important notions of limit of a function, continuity and uniform continuity. The idea of the function f having the limit A at a point c is that the values f(x) are close to A when x is close to c (but different from c). The term 'close to' is explained technically by delta definition.
Continuous functions are very i ......

The notion of compactness is of great importance in analysis especially In connection with continuity. Functions that are continuous on closed and bounded intervals have a number of very important properties that are not possessed by continuous functions on arbitrary sets. Since a closed and bounded interval in R is compact and a set in R is connected if and only if it is an interval. It is import ......

The notion of continuity is one of the important concepts in mathematical analysis. However, not all functions are continuous. If a function fails to be continuous at a point c, then the function is called discontinuous at c, and c is called a point of discontinuity. The set of all points of discontinuity of a function may be a finite , countable , a discrete set, a dense set, or even the entire d ......

The derivative of a function y = f(x) at a point x = c represents the instantaneous rate of change of the function f(x) with respect to the independent variable x at the point c. It is equal to the slope of the tangent to the function at point x = c In other words, dy/dx. = tan theta where theta is the angle between the tangent to the function at x = c and the x axis. The derivative finds applicat ......

The Riemann-Stieltje's integral is the important generalization of the Riemann integral. The Riemann integral is a particular case of a more general integral.

In this unit we shall study the algebraic properties of the Riemann-Stieltje 's integral. i. e. sum, difference and product of two R-S integrable functions are again R-S integrable.

R-S integrals occur in a wide variety of problems but the explicit value of the integrals are obtained in very few cases. In this unit we shall study the mean value theorems which are useful in obtaining estimate value of the R-S integrals.

The concept of bounded variation helps in extending the theories of integration and differentiation. The definition of Riemann-Steiltje's integral can be extended for the cases of monotonically non decreasing functions alpha on [a, b) by employing bounded variation notion. In this unit we shall study some properties of functions of bounded variation.

Sequences of functions are of great importance in many areas of pure and applied mathematics, and their properties can be studied in the context of metric spaces. Sequences of functions are useful in approximating a given function and to define new functions from the
given ones. Point-wise convergence and uniform convergence are of importance when we look at sequences of functions. A type of conv ......

Point wise convergence is not enough to preserve properties of sequences. In other words, if a sequence of functions has some property like continuity(or differentiablility or integrability) and converges point-wise, then the limit function mayor may not have the same property. But uniform continuity is good enough to preserve continuity, but does not preserve differentiability.

In the early nineteenth century most mathematicians believed that a continuous function has derivatives at a significant set of points. But in 1872, in his presentation Berlin academy, Karl Weierstrass shocked the mathematics world by proving this conjecture is false. He presented a function which is continuous everywhere but differentiable nowhere. Because of the contribution of several mathemati ......

In this unit we shall study an important class of series of functions called power series that possess properties that are not valid for general series of function s. At each point interior to the circle of convergence, the power series not only converges but converges absolutely. What is
very important about power series is that, in each circle concentric with the circle of convergence but of sm ......

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