CONTINUITY AND DIFFERENTIABILITY

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5.1 Overview

5.1.1Continuity of a function at a point

Letf be a real function on a subset of the real numbers and letc be a point in the

domain off. Thenf is continuous atc iflim ( ) ( )x cf x f c®=More elaborately, if the left hand limit, right hand limit and the value of the function

atx =c exist and are equal to each other, i.e.,lim ( ) ( ) lim ( ) x cx cf x f c f x-+®®= =thenf is said to be continuous atx =c.

5.1.2Continuity in an interval

(i)f is said to be continuous in an open interval (a,b) if it is continuous at every point in this interval. (ii)f is said to be continuous in the closed interval [a,b] if f is continuous in (a,b) lim x a+® f (x) =f (a) -lim x b® f (x) =f (b)Chapter 5

CONTINUITY AND

DIFFERENTIABILITY

CONTINUITY AND DIFFERENTIABILITY 875.1.3Geometrical meaning of continuity (i) Functionf will be continuous atx =c if there is no break in the graph of the function at the point( ), ( )c f c. (ii) In an interval, function is said to be continuous if there is no break in the graph of the function in the entire interval.

5.1.4Discontinuity

The functionf will be discontinuous atx =a in any of the following cases : (i)lim x a-® f (x) andlim x a+® f (x) exist but are not equal. (ii)lim x a-® f (x) andlim x a+® f (x) exist and are equal but not equal to f (a). (iii) f (a) is not defined.

5.1.5Continuity of some of the common functions

Function f (x)Interval in which

f is continuous

1. The constant function, i.e. f (x) =c

2. The identity function, i.e. f (x) =xR

3. The polynomial function, i.e.

f (x)= a0xn +a1xn-1 + ... +an-1x +an

4. |x -a |(-¥,¥)

5.x-n,n is a positive integer(-¥,¥) - {0}

6.p (x) /q (x), wherep (x) andq (x) areR - {x :q (x) = 0}

polynomials inx

7. sinx, cosxR

8. tanx, secxR- { (2n + 1)π

2:nÎZ}

9. cotx, cosecxR- { (np :nÎZ}

88 MATHEMATICS10.exR

11. logx(0,¥)

12. The inversetrigonometric functions,In their respective

i.e., sin -1x, cos-1x etc.domains

5.1.6Continuity of composite functions

Letf andg be real valued functions such that (fog) is defined ata. Ifg is continuous ata andf is continuous atg(a), then (fog) is continuous ata.

5.1.7Differentiability

The function defined byf¢(x) =0( ) ( )limhf x h f x h

®+ -, wherever the limit exists, is

defined to be the derivative off atx. In other words, we say that a functionf is differentiable at a pointc in its domain if both0( ) ( )lim hf c h f c h -®+ -, called left hand derivative, denoted by L f¢ (c), and0( ) ( )lim hf c h f c h +®+ -, called right hand derivative, denoted by Rf¢ (c), are finite and equal. (i) The functiony =f (x) is said to be differentiable in an open interval (a, b) if it is differentiable at every point of (a, b) (ii) The functiony =f (x) is said to be differentiable in the closed interval [a, b] if Rf¢(a) and Lf¢ (b) exist andf¢ (x) exists for every point of (a,b). (iii) Every differentiable function is continuous, but the converse is not true

5.1.8Algebra of derivatives

Ifu,v are functions ofx, then

(i)( )d u v d x±= ±du dv dx dx(ii)( )= +d dv duuv u vdx dx dx(iii)2du dv v ud udx dx dx v v-

CONTINUITY AND DIFFERENTIABILITY 895.1.9 Chain rule is a rule to differentiate composition of functions. Letf =vou. If

t =u (x) and bothdt dx anddv dt exist then.=df dv dt dx dt dx5.1.10 Following are some of the standard derivatives (in appropriate domains) 1.-1

21(sin )1=-dxdxx2.-1

21(cos )1dxdxx-=-3.-1

21(tan )1=+dxdxx4.-1

21(cot )1dxdxx-=+5.-1

21(sec ), 11dxxdxx x=>-6.-1

21(cosec ), 11dxxdxx x-=>-5.1.11Exponential and logarithmic functions

(i) The exponential function with positive baseb> 1 is the function y =f (x) =bx. Its domain isR, the set of all real numbers and range is the set of all positive real numbers. Exponential function with base 10 is called the common exponential function and with basee is called the natural exponential

5.1 Overview

5.1.1Continuity of a function at a point

Letf be a real function on a subset of the real numbers and letc be a point in the

domain off. Thenf is continuous atc iflim ( ) ( )x cf x f c®=More elaborately, if the left hand limit, right hand limit and the value of the function

atx =c exist and are equal to each other, i.e.,lim ( ) ( ) lim ( ) x cx cf x f c f x-+®®= =thenf is said to be continuous atx =c.

5.1.2Continuity in an interval

(i)f is said to be continuous in an open interval (a,b) if it is continuous at every point in this interval. (ii)f is said to be continuous in the closed interval [a,b] if f is continuous in (a,b) lim x a+® f (x) =f (a) -lim x b® f (x) =f (b)Chapter 5

CONTINUITY AND

DIFFERENTIABILITY

CONTINUITY AND DIFFERENTIABILITY 875.1.3Geometrical meaning of continuity (i) Functionf will be continuous atx =c if there is no break in the graph of the function at the point( ), ( )c f c. (ii) In an interval, function is said to be continuous if there is no break in the graph of the function in the entire interval.

5.1.4Discontinuity

The functionf will be discontinuous atx =a in any of the following cases : (i)lim x a-® f (x) andlim x a+® f (x) exist but are not equal. (ii)lim x a-® f (x) andlim x a+® f (x) exist and are equal but not equal to f (a). (iii) f (a) is not defined.

5.1.5Continuity of some of the common functions

Function f (x)Interval in which

f is continuous

1. The constant function, i.e. f (x) =c

2. The identity function, i.e. f (x) =xR

3. The polynomial function, i.e.

f (x)= a0xn +a1xn-1 + ... +an-1x +an

4. |x -a |(-¥,¥)

5.x-n,n is a positive integer(-¥,¥) - {0}

6.p (x) /q (x), wherep (x) andq (x) areR - {x :q (x) = 0}

polynomials inx

7. sinx, cosxR

8. tanx, secxR- { (2n + 1)π

2:nÎZ}

9. cotx, cosecxR- { (np :nÎZ}

88 MATHEMATICS10.exR

11. logx(0,¥)

12. The inversetrigonometric functions,In their respective

i.e., sin -1x, cos-1x etc.domains

5.1.6Continuity of composite functions

Letf andg be real valued functions such that (fog) is defined ata. Ifg is continuous ata andf is continuous atg(a), then (fog) is continuous ata.

5.1.7Differentiability

The function defined byf¢(x) =0( ) ( )limhf x h f x h

®+ -, wherever the limit exists, is

defined to be the derivative off atx. In other words, we say that a functionf is differentiable at a pointc in its domain if both0( ) ( )lim hf c h f c h -®+ -, called left hand derivative, denoted by L f¢ (c), and0( ) ( )lim hf c h f c h +®+ -, called right hand derivative, denoted by Rf¢ (c), are finite and equal. (i) The functiony =f (x) is said to be differentiable in an open interval (a, b) if it is differentiable at every point of (a, b) (ii) The functiony =f (x) is said to be differentiable in the closed interval [a, b] if Rf¢(a) and Lf¢ (b) exist andf¢ (x) exists for every point of (a,b). (iii) Every differentiable function is continuous, but the converse is not true

5.1.8Algebra of derivatives

Ifu,v are functions ofx, then

(i)( )d u v d x±= ±du dv dx dx(ii)( )= +d dv duuv u vdx dx dx(iii)2du dv v ud udx dx dx v v-

CONTINUITY AND DIFFERENTIABILITY 895.1.9 Chain rule is a rule to differentiate composition of functions. Letf =vou. If

t =u (x) and bothdt dx anddv dt exist then.=df dv dt dx dt dx5.1.10 Following are some of the standard derivatives (in appropriate domains) 1.-1

21(sin )1=-dxdxx2.-1

21(cos )1dxdxx-=-3.-1

21(tan )1=+dxdxx4.-1

21(cot )1dxdxx-=+5.-1

21(sec ), 11dxxdxx x=>-6.-1

21(cosec ), 11dxxdxx x-=>-5.1.11Exponential and logarithmic functions

(i) The exponential function with positive baseb> 1 is the function y =f (x) =bx. Its domain isR, the set of all real numbers and range is the set of all positive real numbers. Exponential function with base 10 is called the common exponential function and with basee is called the natural exponential