## Monday, August 13, 2012

### Calculus - Limits and Continuity

The concept of Limitlays the foundation for the popular branch of Mathematics called Calculus. Calculusinvolves the analysis of functions and their behaviour. To study the behaviour of functions one needs to have a good hold on the fundamental concepts of Limit and Continuity. Calculus Limits and Continuity is the language of science and engineering.Limits and Continuity in Calculus have led to the development of ideas like derivative, integration etc.

Consider a function f(x):R->R and ‘a’ is any point in the domain of the function, limit of the function f(x) exists at x = a, if f(x) = f(a), where x is arbitrarily close to a. The idea is that as we approach the point x = a on the real line, f(x) approaches f (a). The limiting value of the function when x is very close to ‘a’ is ‘f (a)’. In the language of mathematics it is commented as “lim x->a, f(x) =f(a)”.

Continuity is another important concept which is based on the concept of limits. The function f(x) is said to be continuous at x = a, if the limit of the function at x = a on approaching the point x = a from both sides is equal to the value of the function at x = a. Precisely this can be stated as – if  lim h->0, f(a+h) = f(a-h)=f(a), then f(x) is continuous at x = a. Mathematically this can be stated as – a function is said to be continuous at a point if the right hand limit and the left hand limit of the function at that point is equal to the value of the function at that point. Geometrically speaking, a continuousfunction is the one which can be sketched on paper without lifting the pen even once.

Properties of Continuous Functions:There are two important results for continuous functions which are stated in the form of theorems-
Intermediate value theorem: The intermediate value theoremstates that if f is a real valued continuousfunctionon the closed interval [a, b] andt is any number between f(a) and f(b), then there exists a numberc in [a, b] such that f(c) = t.Example: The height of a child increases from 1 m to 1.7 m between the ages of eight and sixteen years, then, at some point of timebetween eight and sixteen years of age, the child must have had a height of 1.5 m.
Extreme value theorem: The extreme value theorem states that if f is continuousreal valued function on [a,b], then f has a maximumvalue in [a,b], i.e. there exists some c in [a,b] such that f(c) >= f(x)  for all x in  [a,b]. The same holds for the minimum of f(x). For example, consider the function sin(x) where x lies between [0,2*pi]. This function attains at maximum at x = pi/2 and a minimum at x = 3*pi/2.