Paper 4: Fundamentals of Business Mathematics & Statistic

3.18 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

If however all these values are equal, then f (x) is continuous at x = a, otherwise it is discontinuous.

Example 44 : Show that f(x) = 3x^2 – x + 2 is continuous at x = 1.

Solution:

Now, x 1limf(x) lim 3x x 2 lim 3 1 h 1 h 2→ + =x 1→ +(^2 − + )=h 0→ { (+ )^2 − +( )+ }

[Putting x = 1 + h as x → 1, h → 0]

= h 0lim 3h 5h 4 3.0 5.0 4 4→(^2 + + )= + + =

and h 1limf x lim 3x x 2 lim 3 1 h 1 h 2→ −( )=x 1→ −(^2 − + )=h 0→ { (− )^2 − −( )+ }

(x = 1 – h, h → 0 as x → 1)

h 0lim 3h 5h 4 3.0 5.0 4 4→ (^2 − + )= − + =

Again f (1) = 3.1^2 – 1 + 2 = 4

Thus we find that all the values are equal.

∴ f (x) is continuous at x = 1 ;

εεεεε - δδδδδ Definition :

Again corresponding to definition of limit, we may define the continuity of a function as follows :

The functions f(x) is continuous at x = 1, if f (a) exists and for any pre-assigned positive quantity ε, however

small we can determine a positive quantity δ, such that |f (x) – f (a) |< ε, for all values of x satisfying

|x – a | < δ.

Some Properties :

1. The sum or difference of two continuous functions is a continuous function
   (i.e.,) x alim f x x limf x lim x .→ {( )± φ( )}=x a→ ( )±x a→ φ( )


2. Product of two continuous functions is a continuous function.


3. Ratio of two continuous functions is a continuous function, provided the denominator is not zero.
   Continuity in an Interval, at the End Points :
   A function is said to be continuous over the interval (open or closed) including the end points if it is continuous
   at every point of the same interval.
   Let c be any point in the interval (a, b) and if x clim f x lim f x f c ,→ + ( )=x c→ − ( )= ( )then f(x) is continuous in the interval
   (a, b)

A function f(x) is said to continuous at the left end of an interval a ≤ x ≤ b (^) x alim f x f a ,→ + ( )= ( ) if and at the right
end b if x blim f x f b .→ − ( )= ( )
Discontinuity at a Point : If at any point x = a in its domain, at least one of values x alim f x , lim f x→ + ( ) x a→ −( ) and f(a)
be different from the others, then f(x) fails to be continuous at that point, i.e., at x = a.