1. Prove that the function f (x) = 5x 3 is continuous at x = 0, at x = 3 and at x = 5. 2. Examine the continuity of the function f (x) = 2x2 1 at x = 3. 3. Examine the following functions for continuity. (a) f (x) = x 5 (b) f (x) = 1 x 5 , x 5 (c) f (x) = 2 25 5 x x + , x 5 (d) f (x) = | x 5| 4. Prove that the function f (x) = xn is continuous at x = n, where n is a positive integer. 5. Is the function f defined by , if 1 ( ) 5, if >1 x x f x x = continuous at x = 0? At x = 1? At x = 2? Find all points of discontinuity of f, where f is defined by 6. 2 3, if 2 ( ) 2 3, if > 2 x x f x x x + = 7. | | 3, if 3 ( ) 2 , if 3 < 3 6 2, if 3 x x f x x x x x + = < + 8. | | , if 0 ( ) 0, if 0 x x f x x x = = 9. , if 0 ( ) | | 1, if 0 x x f x x x < = 10. 2
Question 1:Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Question 2:Examine the continuity of the function f (x) = 2x2 – 1 at x = 3.
Question 3:Examine the following functions for continuity. (a) f (x) = x – 5 (b) f (x) = 1 x − 5 , x ¹ 5 (c) f (x) = 2 25 5 x x − + , x ¹ –5 (d) f (x) = | x – 5|
Question 4:Prove that the function f (x) = xn is continuous at x = n, where n is a positive integer.
Question 5:Is the function f defined by , if 1 ( ) 5, if >1 x x f x x ≤ = continuous at x = 0? At x = 1? At x = 2? Find all points of discontinuity of f, where f is defined by
Question 6:2 3, if 2 ( ) 2 3, if > 2 x x f x x x + ≤ = −
Question 7:| | 3, if 3 ( ) 2 , if 3 < 3 6 2, if 3 x x f x x x x x + ≤ − = − − < + ³
Question 8:| | , if 0 ( ) 0, if 0 x x f x x x ¹ = =
Question 9:, if 0 ( ) | | 1, if 0 x x f x x x < = − ³
Question 10:2 1, if 1 ( ) 1, if 1 x x f x x x + ³ = + <
Question 11:3 2 3, if 2 ( ) 1, if 2 x x f x x x − ≤ = + >
Question 12:10 2 1, if 1 ( ) , if 1 x x f x x x − ≤ = >
Question 13:Is the function defined by 5, if 1 ( ) 5, if 1 x x f x x x + ≤ = − > a continuous function? Discuss the continuity of the function f, where f is defined by
Question 14:3, if 0 1 ( ) 4, if 1 3 5, if 3 10 x f x x x ≤ ≤ = < < ≤ ≤
Question 15:2 , if 0 ( ) 0, if 0 1 4 , if >1 x x f x x x x < = ≤ ≤
Question 16: 2, if 1 ( ) 2 , if 1 1 2, if 1 x f x x x x − ≤ − = − < ≤ >
Question 17:Find the relationship between a and b so that the function f defined by 1, if 3 ( ) 3, if 3 ax x f x bx x + ≤ = + > is continuous at x = 3.
Question 18:For what value of l is the function defined by 2 ( 2 ), if 0 ( ) 4 1, if 0 x x x f x x x l − ≤ = + > continuous at x = 0? What about continuity at x = 1?
Question 19: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.
Question 20:Is the function defined by f (x) = x2 – sin x + 5 continuous at x = p?
Question 21: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
Question 22:Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Question 23:Find all points of discontinuity of f, where sin , if 0 ( ) 1, if 0 x x f x x x x < = + ³
Question 24:Determine if f defined by 2 1 sin , if 0 ( ) 0, if 0 x x f x x x ¹ = = is a continuous function?
Question 25:Examine the continuity of f, where f is defined by sin cos , if 0 ( ) 1, if 0 x x x f x x − ¹ = − = Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29
Question 26: cos , if 2 2 ( ) 3, if 2 k x x x f x x p ¹ π − = π = at x = 2 p
Question 27: 2 , if 2 ( ) 3, if 2 kx x f x x ≤ = > at x = 2
Question 28: 1, if ( ) cos , if kx x f x x x + ≤ p = > p at x = p
Question 29: 1, if 5 ( ) 3 5, if 5 kx x f x x x + ≤ = − > at x = 5
Question 30:Find the values of a and b such that the function defined by 5, if 2 ( ) , if 2 10 21, if 10 x f x ax b x x ≤ = + < < ³ is a continuous function.
Question 31:Show that the function defined by f (x) = cos (x2) is a continuous function.
Question 32:Show that the function defined by f (x) = | cos x| is a continuous function.
Question 33:Examine that sin | x| is a continuous function.
Question 34:Find all the points of discontinuity of f defined by f (x) = | x| – | x + 1|.