T4Q4 Show that the function F C0 1 rightarrow C0 1 given by

T4Q4

Show that the function F: C[0, 1) rightarrow C[0, 1] given by F(f)(x) = cos f(x) for each f C[0, 1] is (d_max, d_max)-continuous on C[0, 1]. You may find it helpful to use the fact that |cos a - cosb| lessthanorequalto |a - b| for a, b R.

Solution

Here the function values limit from 0 to 1

Here the fact | cos a – cos b| |a-b| for all a,b€R

It is true for an example Take some random values like a= 60, b=90

        | cos60 – cos90| | 45-90|

        | ½ -0| |-45|

        0.5 45 à like this for all the values it will become true

The cosine function has a values from 0 to 1 the domain of the function is true

C [0, 1] is a valid domain for this function.

Minimum value of Cosine function is 0 for all Real numbers

Maximum value of the Cosine function is 1 for real numbers

                                   

F (f(x)) = cos f(x)

For different values 30, 45, 60, 90,180

F(f(x)) = cos 30

            =   à it is less than zero

For 60 is   = cos 60

                  = ½

For 90 is = cos 90

                =0

For 0 is = cos 0

              =1

From the above calculation it is clearly shows that F(f(x))= cos f(x)

It will always varies from the 0 to 1