Continuity

CONTINUITY

A function is said to be continuous over an interval if it is continuous at each point an that interval. Graphically, a function is continuous when its graph can be drawn without lifting the pen off the paper.

Mathematically, A function f(x) is said to be continuous at a point x = a if

limit _xèa f(x) = f(a)

From the above definition, we can conclude that if the function f is continuous at a point x = a, then

the function should be defined at x = a i.e. f(a) should exist.

the limit of the function, limit _xèaf(x) should exist (i.e. left and right hand limits should be equal).

the two should two should be equal.

If the limit exists but is not equal to f(a) (this also includes the case when f(a) is not defined), then we say that the function has a Removable Discontinuity at x = a.

The discontinuity is called ‘removeable’ as the function can be made continuous by putting

f(a) = value of the limit limit _xèa f(x).

If, on the other hand, limit _xèaf(x) does not exist (i.e. left and right hand limits are not equal) then the function is said to have a non-removeable discontinuity.

NOTE: The functions sinx, cosx, a^x, logx and all polynomials in x and all continuous over their domains.

Basic properties of limits imply the following:

Theorem. If f(x) and g(x) are continuous at a. Then

(1)

f(x) + g(x) is continuous at a;

(2)

$\alpha f(x)$ is continuous at a, where $\alpha$ is an arbitrary number;