REVISION - XII

CONTINUITY & DIFFERENTIABLE 

1) Show that f(x)= x³ is continuous at x= 2.

2) Show that f(x)=[ x] is not continuous at x= n, where n is any integer.

3) Show that f(x)={ x,   if x is an integer 

                                  0,  if x is not integer 

 is discontinuous at each integral value of x.

4) Show that f(x)={x/|x|, when x ≠ 0,

                                 1, when x= 0

is discontinuous at x= 0.

5) If f(x)={ (x²-1)/(x -1) for x≠ 1

                    2 for x= 1

Show that f(x) is continuous at x= 1.

6) Determine the value of k for which the function 

f(x)={   (sin5x)/3x, if x≠ 0

                  k,          if x= 0

is continuous at x=0.           5/3

7) Show that the function is continuous at x= 0

f(x)={ x sin(1/x), when x ≠ 0

             0,              when x = 0

8) Let f(x)={ (sinx)/x + cosx, when x ≠ 0

                        2, when x= 0

Show that f(x) is continuous at x= 0.

9) Show that the function is discontinuous at x= 0

f(x)={ (sin²ax)/x, when x≠ 0

             1,              when x= 0

Redefine the function in such a way that it becomes continuous at x= a.

10) Is the function, f(x)= (3x + 4 tanx)/x continuous at x= 0 ? If not, how many the function be defined to make it continuous at this point ?        No, 

11) Discuss the continuity of the function 

f(x)={ 3x -2, when x≤ 0

            x +1, when x> 0 at x= 0.

12) Discuss the continuity of the function 

f(x)={ - x, when x≤ 0

            x, when 0< x ≤1

         2-x, when 1< x<2

            1, when x> 2

at each of the point x= 0,1,2.       Y:0, y:1, n:2

13) Show that the function 

f(x)={ 2x, if x< 2

           2, if x= 2

           x², if x> 2

has a removable discontinuity at x= 2.

CONTINUOUS FUNCTIONS

1) Let f(x)={ x if x≥ 1

                     x² if x< 1

Is a continuous function? Why ?

2) Prove that f(x)= |x | is a continuous function.

3) Discuss the continuity of the function.

f(x)= {2x -1, if x< 0;

          2x +1, if x≥ 0.

4) Discuss the continuity of the function 

f(x)= { (sinx)/x, if x < 0;

            (x+1), if x≥ 0.

5) Discuss the continuity of the function 

f(x)={ x/|x|, if x ≠ 0;

             0, if x= 0.

6) Locate the point of discontinuity of the function 

f(x)= { (x⁴-16)/(x -2), if x≠ 2

               16, if x= 2.

7) Determine the value of k so that the function 

f(x)={ kx², if x< 2;

           3, if x> 2 is continuous.

8) Let f(x)={ 1, if x≤ 3;

                    ax+ b, if 3<x<5;

                      7, if 5≤ x.

Find the values of a and b so that f(x) is continuous.

9) Show that the function f(x)= √(x⁴+3) is continuous at each point.

10) Show that the function f(x)= |sinx + cosx| is continuous at x=π.

DIFFERENTIABLE