Exercise 5.1

Prove that the function f (x) = 5x - 3 is continuous at x = 0, at x = - 3 and at x = 5.

Examine the continuity of the function f (x) = 2x2 - 1 at x = 3.

Examine the following functions for continuity. (a) f (x) = x - 5 (b) f (x) = 1 5x − , x ≠ 5 (c) f (x) = / 2 25 / x / x − + , x ≠ -5 (d) f (x) = | x - 5 |

Prove that the function f (x) = x n is continuous at x = n, where n is a positive integer.

Is the function f defined by , if 1( ) 5, if > 1

≤=   continuous at x = 0? At x = 1? At x = 2? Find all points of discontinuity of f, where f is defined by

2 3, if 2( ) 2 3, if > 2

• ≤=  − 7. | | 3, if 3

( ) 2 , if 3 < 3 6 2, if 3 x x

x  ≠=   =

, if 0| |( ) 1, if 0 x xxf x x  <=  − ≥ 10. 2 1, if 1

( ) 1, if 1 x x f x x x + ≥=  + <

3/2 3, if 2( ) 1, if 2 x xf x x x  − ≤=  + >

10/2 1, if 1( ) , if 1 x xf x x x  − ≤=  >

Is the function defined by 5, if 1( ) 5, if 1

• ≤=  − > a continuous function? Discuss the continuity of the function f, where f is defined by 14. 3, if 0 1 ( ) 4, if 1 3 5, if 3 10 x | f | x | x | |---|---|---|

x ≤ ≤ = < <  ≤ ≤ 15. 2 , if 0 ( ) 0, if 0 1 4 , if > 1 x x

x x / < = ≤ ≤  16. 2, if 1 ( ) 2 , if 1 1 2, if 1 x

x / − ≤ − = − < ≤  >

Find the relationship between a and b so that the function f defined by 1, if 3( ) 3, if 3 ax xf x bx x + ≤=  + > is continuous at x = 3.

For what value of λ is the function defined by 2( 2 ), if 0( ) 4 1, if 0 x x xf x x x λ − ≤=  + > continuous at x = 0? What about continuity at x = 1?

Show that the function defined by g (x) = x - [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Is the function defined by f (x) = x2 - sin x + 5 continuous at x = π?

Discuss the continuity of the following functions: (a) f (x) = sin x + cos x (b) f (x) = sin x - cos x (c) f (x) = sin x . cos x

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Find all points of discontinuity of f, where sin , if 0( ) 1, if 0

x x  <=   + ≥

Determine if f defined by 2 1sin , if 0( ) 0, if 0

x  ≠=   = is a continuous function? MATHEMATICS118

Examine the continuity of f, where f is defined by sin cos , if 0( ) 1, if 0

− ≠= − = Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29. 26. cos , if2 2( ) 3, if 2 k x xxf x x π ≠ π −=  π = at x = 2 π 27. 2, if 2( ) 3, if 2 kx xf x x  ≤=  > at x = 2 28. 1, if( ) cos , if

• ≤ π=  > π at x = π 29. 1, if 5( ) 3 5, if 5 | kx xf | x | x | x | |---|---|---|---|

• ≤=  − > at x = 5 30. Find the values of a and b such that the function defined by 5, if 2 | ( ) , if | 2 | 10 | |---|---|---|

21, if 10 x

( ) ( )lim h

( ) ( )( ) lim h f x h f xf x h→ + −′ = wherever the limit exists is defined to be the derivative of f. The derivative of f is denoted by f ′ (x) or ( ( ))d f xdx or if y = f (x) by dy dx or y′. The process of finding derivative of a function is called differentiation. We also use the phrase differentiate f (x) with respect to x to mean find f ′(x). The following rules were established as a part of algebra of derivatives: (1) (u ± v)′ = u′ ± v′ (2) (uv)′ = u′v + uv′ (Leibnitz or product rule) (3) u u v uv v v ′ ′ − ′  =   , wherever v ≠ 0 (Quotient rule). The following table gives a list of derivatives of certain standard functions: Table 5.3 Whenever we defined derivative, we had put a caution provided the limit exists. Now the natural question is; what if it doesn’t? The question is quite pertinent and so is its answer. If ( ) ( )lim h

h→ + − does not exist, we say that f is not differentiable at c. In other words, we say that a function f is differentiable at a point c in its domain if both -0 ( ) ( )lim h

h→ + − and ( ) ( )lim h

h / +→ + − are finite and equal. A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b]. As in case of continuity, at the end points a and b, we take the right hand limit and left hand limit, which are nothing but left hand derivative and right hand derivative of the function at a and b respectively. Similarly, a function is said to be differentiable in an interval (a, b) if it is differentiable at every point of (a, b). MATHEMATICS120