# Continuity of functions-

The word continuous means without any break or gap. Continuity of functions exists when our function is without any break or gap or jump . If there is any gap in the graph, the function is said to be discontinuous.

Graph of functions like sinx,cosx, secx, 1/x etc are continuous (without any gap) while greatest integer function has a break at every point(discontinuous).

1. A function f(x) is said to be continuous at x = c,  if  .

symbolically f is continuous at x = c if .

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighborhood of x = a, not necessarily at x = a.

2.Reasons of discontinuity:

(i)   does not exist   i.e.

(ii)  f(x) is not defined at x= c

(iii)

3.Types of Discontinuities

Removable type of discontinuities- In this case exists but is not equal to f(c) then the function is said to have a removable discontinuity or discontinuity of the first kind. In this case we can redefine the function such that f(x) = f(c) & make it continuous at x= c.

Removable type of discontinuity can be further classified as:

(a) Missing Point Discontinuity- Where f(x) exists finitely but f(a) is not defined.e.g.

here f(x) has a missing point discontinuity at x = 1 , and

has a missing point discontinuity at x = 0

(b) Isolated Point Discontinuity- Where f(x) exists & f(a) also exists but

e.g.  here    & f (4) = 9 has an isolated point

discontinuity at x = 4. Similarly f(x) = [x] + [ –x] =   has an isolated point discontinuity at all xI.

Non-Removable type of discontinuities- In case f(x) does not exist then it is not possible to make the function continuous by redefining it. Such discontinuities are known as non-removable discontinuity or discontinuity of the 2nd kind.

Non-removable type of discontinuity can be further classified as:

(a) Finite discontinuity- e.g. f(x) = x – [x] at all integral,

f(x) = at x = 0 and       at x = 0  [note that f(0+) = 0 ; f(0) = 1]

(b) Infinite discontinuity-  e.g.   or    at x = 4,

at x =0 and    x = 0.

(c) Oscillatory discontinuity- e.g.    at x = 0. In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but    does not exist.

4.The Jump Of Discontinuity- In case of discontinuity of the second kind the non-negative difference between the value of the RHL at x = c & LHL at x = c is called The Jump Of Discontinuity. A function having a finite number of jumps in a given interval I is called a Piece Wise Continuous or Sectionally Continuous function in this interval.
5. All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains.

6. If f & g are two functions that are continuous at x= c then the functions defined by :

F1(x) = f(x) ± g(x)  ;  F2(x) = K f(x) , K any real number  ; F3(x) = f(x).g(x) are also continuous at x= c. Further, if g (c) is not zero, then    is also continuous at   x= c.

Theorems of Continuity-

(a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function is not necessarily be discontinuous at x = a. e.g. f(x) = x &

g(x) =

(b) If f(x) and g(x) both are discontinuous at x = a then the product function

is not necessarily be discontinuous at x = a. e.g

(c) Point functions are to be treated as discontinuous. eg.   is not continuous at x = 1.

(d) A Continuous function whose domain is closed must have a range also in closed interval.

(e) If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is

continuous at x = c. eg.     &      are continuous at x = 0 , hence the

composite      will also be continuous at x = 0

(f)  this nth degree polynomial is continuous for xR

(g)  y=Sinx, y=Cosx are continuous  for xR

(h)  y=  is continuous  for all x>0

(i)  y=  is continuous  for all   xR

7. Continuity In An Interval-
(a) A function f is said to be continuous in (a, b) if f is continuous at each & every point     Î(a, b)

(b) A function f is said to be continuous in a closed interval [a,b] if:

(i) f is continuous in the open interval (a, b) &

(ii) f is right continuous at ‘a’ i.e.   = a finite quantity.

(iii) f is left continuous at ‘b’ i.e.    = a finite quantity. Note  that a function f which is continuous in possesses the following properties :

Note  that a function f which is continuous in [a,b] possesses the following properties:

(i) If f(a) & f(b) possess opposite signs, then there exists at least one solution of the equation f(x) = 0 in the open interval (a , b).

(ii) If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open interval (a, b).

(ii) If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open interval (a, b).

Sandwich Theorem or Squeeze Theorem-   Suppose         in some interval about c and that f(x) and h(x) approaches the same limit L as approaches c i.e

then

This theorem is known as Sandwich Theorem.

Intermediate Value Theorem- If we have a function f(x) that is continuous in the closed interval [a,b] and we suppose M number  between f(a) and f(b) then there exists a number c in such a way that-

(i)

(ii)   f(c)=M

Example 1  Given the graph of f(x), shown below, determine if f(x) is continuous at , and .

Solution        First x=-2

The function value and the limit are different so the function is discontinuous at this point.  This type of discontinuity in a graph is considere as  jump discontinuity. It occurs where the graph has a break in it.

Now at x=0

So the function is continuous at this given point.

At x=3

Clearly, the function is discontinuous here. Graph is exihibiting a whole at given point. we call it removable discontinuity

Example-2  Check the continuity of the below function

Ans- Rational function can be discontinuous if its denominator is zero

so the given function is continuous everywhere except x=5 and x=-3

Here I am posting practice problems on ‘limits and continuity’ as well as a module on the same topics

### practice problems limits and continuity.pdf

#### Post on limits-part two

• Very useful information, we IB teacher required ,this type of information for all the concepts Mathematics.

• Thank you very much, sir
please suggest how can we make it more useful for our students?

• sandeep kumar

Si, please sand 11th or 12th maths paper level of iit advance and maind

• Which year?