CONTINUITY AND DIFFERENTIABILITY

213507

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 function. (ii) Letb > 1 be a real number. Then we say logarithm ofa to baseb isx ifbx=a, Logarithm ofa to the baseb is denoted by logba. If the baseb = 10, we say it is common logarithm and ifb = e, then we say it is natural logarithms. logx denotes the logarithm function to basee. The domain of logarithm function isR+, the set of all positive real numbers and the range is the set of all real numbers. (iii) The properties of logarithmic function to any baseb> 1 are listed below:

1. log

b (xy) = logbx + logby

2. log

y = logbx - logby

90 MATHEMATICS3. logbxn = nlogb x

4.logloglogc

b cx xb= , wherec > 1

5. log

bx1 log= xb6. logbb =1 and logb 1 = 0 (iv) The derivative ofexw.r.t.,x isex , i.e.( )x xde edx=. The derivative of logx w.r.t.,x is1 x; i.e.1(log )dxdx x=.

5.1.12Logarithmic differentiation is a powerful technique to differentiate functions

of the formf (x) = (u (x))v(x), where bothf andu need to be positive functions for this technique to make sense.

5.1.13Differentiation of a function with respect to another function

Letu =f (x) andv =g (x) be two functions ofx, then to find derivative of f (x) w.r.t. tog (x), i.e., to finddu dv, we use the formuladu dudx dvdv dx=.

5.1.14Second order derivative2

2d dy d y

dx dx y

2 , ify =f (x).

5.1.15Rolle"s Theorem

Letf : [a,b]®R be continuous on [a,b] and differentiable on (a,b), such thatf(a) =f (b), wherea andb are some real numbers. Then there exists at least one pointc in (a,b) such thatf¢ (c) = 0.

CONTINUITY AND DIFFERENTIABILITY 91Geometrically Rolle"s theorem ensures that there is at least one point on the curve

y =f (x) at which tangent is parallel tox-axis (abscissa of the point lying in (a,b)).

5.1.16Mean Value Theorem (Lagrange)

Letf : [a,b]®R be a continuous function on [a,b] and differentiable on (a,b). Then there exists at least one pointc in (a,b) such thatf¢ (c) =( ) ( )f b f a b a- Geometrically, Mean Value Theorem states that there exists at least one pointc in (a,b) such that the tangent at the point (c,f (c)) is parallel to the secant joining the points (a,f(a) and (b,f (b)).

5.2 Solved Examples

Short Answer (S.A.)

Example 1Find the value of the constantk so that the functionf defined below is continuous atx = 0, where21- cos4( ), 08 , 0x f xxx k xì SolutionIt is given that the functionf is continuous atx = 0. Therefore,0limx®f(x) =f(0)

Þ201-cos4lim8xxkx®=Þ2

2

02sin 2lim8xxkx®=Þ2

Thus,f is continuous atx = 0 ifk = 1.

Example 2Discuss the continuity of the functionf(x) = sinx . cosx. SolutionSince sinxand cosx are continuous functions and product of two continuous function is a continuous function, thereforef(x) = sinx. cosxis a continuous function.

92 MATHEMATICSExample 3 If3 2

2-16 20, 2( )( - 2)

, 2x x x

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 function. (ii) Letb > 1 be a real number. Then we say logarithm ofa to baseb isx ifbx=a, Logarithm ofa to the baseb is denoted by logba. If the baseb = 10, we say it is common logarithm and ifb = e, then we say it is natural logarithms. logx denotes the logarithm function to basee. The domain of logarithm function isR+, the set of all positive real numbers and the range is the set of all real numbers. (iii) The properties of logarithmic function to any baseb> 1 are listed below:

1. log

b (xy) = logbx + logby

2. log

y = logbx - logby

90 MATHEMATICS3. logbxn = nlogb x

4.logloglogc

b cx xb= , wherec > 1

5. log

bx1 log= xb6. logbb =1 and logb 1 = 0 (iv) The derivative ofexw.r.t.,x isex , i.e.( )x xde edx=. The derivative of logx w.r.t.,x is1 x; i.e.1(log )dxdx x=.

5.1.12Logarithmic differentiation is a powerful technique to differentiate functions

of the formf (x) = (u (x))v(x), where bothf andu need to be positive functions for this technique to make sense.

5.1.13Differentiation of a function with respect to another function

Letu =f (x) andv =g (x) be two functions ofx, then to find derivative of f (x) w.r.t. tog (x), i.e., to finddu dv, we use the formuladu dudx dvdv dx=.

5.1.14Second order derivative2

2d dy d y

dx dx y

2 , ify =f (x).

5.1.15Rolle"s Theorem

Letf : [a,b]®R be continuous on [a,b] and differentiable on (a,b), such thatf(a) =f (b), wherea andb are some real numbers. Then there exists at least one pointc in (a,b) such thatf¢ (c) = 0.

CONTINUITY AND DIFFERENTIABILITY 91Geometrically Rolle"s theorem ensures that there is at least one point on the curve

y =f (x) at which tangent is parallel tox-axis (abscissa of the point lying in (a,b)).

5.1.16Mean Value Theorem (Lagrange)

Letf : [a,b]®R be a continuous function on [a,b] and differentiable on (a,b). Then there exists at least one pointc in (a,b) such thatf¢ (c) =( ) ( )f b f a b a- Geometrically, Mean Value Theorem states that there exists at least one pointc in (a,b) such that the tangent at the point (c,f (c)) is parallel to the secant joining the points (a,f(a) and (b,f (b)).

5.2 Solved Examples

Short Answer (S.A.)

Example 1Find the value of the constantk so that the functionf defined below is continuous atx = 0, where21- cos4( ), 08 , 0x f xxx k xì SolutionIt is given that the functionf is continuous atx = 0. Therefore,0limx®f(x) =f(0)

Þ201-cos4lim8xxkx®=Þ2

2

02sin 2lim8xxkx®=Þ2

Thus,f is continuous atx = 0 ifk = 1.

Example 2Discuss the continuity of the functionf(x) = sinx . cosx. SolutionSince sinxand cosx are continuous functions and product of two continuous function is a continuous function, thereforef(x) = sinx. cosxis a continuous function.

92 MATHEMATICSExample 3 If3 2

2-16 20, 2( )( - 2)

, 2x x x