# HOW TO EXAMINE THE CONTINUITY OF THE FUNCTION

## About "How to Check Continuity of a Function If Interval is not Given"

How to Check Continuity of a Function If Interval is not Given :

Here we are going to how to examine the continuity of the function when the interval is not given.

Three requirements have to be satisfied for the continuity of a function y = f(x) at x = x0 :

(i) f(x) must be defined in a neighbourhood of x0 (i.e., f(x0) exists);

(ii) lim x->xf(x) exists.

(iii) f(x0)  =  lim x -> x0 f(x)

## Points to Remember While Checking Continuity of a Functions

(1)  Constant function is continuous at each point of R (R stands for real numbers)

(2)  Power functions with positive integer exponents are continuous at every point of R

(3)  Polynomial functions, p(x) are continuous at every point of R.

(4)  Quotients of polynomials namely rational functions of the form

R(x) = p(x) / q(x), are continuous at every point where q(x) ≠ 0, and

(5)  The circular functions sin x and cos x are continuous at every point of their domain  R  = (- ∞, ∞) since lim x-> x0 sin x = sin x0, lim x-> x0 cos x  =  cos x0

(6)  The nth root functions, f (x) = x1/n are continuous in their proper domain since lim x -> x0 (x1/n)  =  x0 1/n

(7)  The reciprocal function f (x) = 1/x is not defined at 0 and hence it is not continuous at 0. It is continuous at each point of R − {0} .

(8)  The exponential function f(x) = ex is continuous on R.

(9) The logarithmic function f(x) = log x (x > 0) in continuous in (0, ∞)

(10)  The modulus function

f(x) = |x|

-x if x < 0

0 if x = 0

x if x > 0

is continuous at all points of the real line R .

Question 1 :

Prove that f(x) = 2x2 + 3x - 5 is continuous at all points in R.

Solution :

Since the given function f(x) is a polynomial, it is continuous at all points in R.

Question 2 :

Examine the continuity of the following

x + sin x

Solution :

Let f(x)  =  x + sin x

(i)  From the given function, we come to know that both "x" and "sin x" are defined for all real numbers.

(ii)  lim x-> x0 f(x)  =  lim x-> x0 x + sin x

By applying the limit, we get

=  x0 + sin x0 -------(1)

(iii) f(x0)  =  x0 + sin x0 -------(2)

From (1) and (2)

lim x-> x0 f(x)  =  f(x0)

Hence the given function is continuous for all real values.

After having gone through the stuff given above, we hope that the students would have understood, "How to Check Continuity of a Function If Interval is not Given"

Apart from the stuff given in "How to Check Continuity of a Function If Interval is not Given" if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6