Continuity and Differentiability

Let f(x) = 0 ( x - x ( 2 ) (x - 1)1 + x ( 2 )(x - 1) ), x R. Consider the following two statements : (I) f(x) is discontinuous at x = 1. (II) f(x) is continuous at x = -1. Then,

1. Neither (I) nor (II) is True
2. Only (II) is True
3. Only (I) is True
4. Both (I) and (II) are True

Let , R be such that the function f(x) = cases 2 (x2 - 2 ) + 2 x, & x < 1 \ ( + 3)x + ( - ), & x 1 cases be differentiable at all x R. Then 34( + ) is equal to

1. 48
2. 84
3. 36
4. 24

Let [t] denote the greatest integer less than or equal to t. If the function f(x) = cases b2 ( 2 [ 2( x + x) x ] ), & x < 0 \ x - 12 2xx3, & x > 0 \ a, & x = 0 cases is continuous at x = 0, then a2 + b2 is equal to :

1. 12
2. 58
3. 34
4. 916

Let f(x) = cases ax2 + 2ax + 34x2 + 4x - 3 & , x - 32, 12 \ b & , x = - 32, 12 cases be continuous at x = - 32. If f f(x) = 75, then x is equal to:

1. 1.4
2. 2
3. 1
4. 0

If f(x) = cases a|x| + x2 - 2( |x|)( |x|)x & , x 0 \ b & , x = 0 cases is continuous at x = 0, then a + b is equal to :

1. 1
2. 2
3. 0
4. 4

Let [ ] denote the greatest integer function, and let f(x) = \ 2x, x2 \. Let S = \x (-2, 2) : the function g(x) = |x| [x2 ] is discontinuous at x\. Then x S f(x) equals

1. 2 - 2
2. 2 6 - 3 2
3. 1 - 2
4. 6 - 2 2

The number of points of discontinuity of the function f(x) = [ x22 ] - [ x], x [0, 4], where [ ] denotes the greatest integer function, is .

If the function f(x) = ( x) - ( x) x - x is continuous at x = 0, then f(0) is equal to .

Let m and n be the number of points at which the function f(x) = \x, x3, x5, , x21 \, x R, is not differentiable and not continuous, respectively. Then m + n is equal to .

Let f(x) = cases (1 + ax)1/x, & x < 0 \ 1 + b, & x = 0 \ (x+4)1/2 - 2(x+c)1/3 - 2, & x > 0 cases be continuous at x = 0. Then ea bc is equal to:

1. 64
2. 48
3. 36
4. 72