What is the Total Variation of a Function?

[Eren10]

hi,

I have to calculate total variation of this function:

1 for x< 0
sin(pi * x) for 0<= x <= 3
2 for x> 3

I could not find any example for doing this. Can someone help me ?

 

[mathman]

It is straightfoward: at x=0, the function jumps from 1 to 0 (var = 1); from 0 to pi/2, it goes from 0 up to 1 (var = 1); from pi/2 to 3pi/2, it goes down from 1 to -1 (var = 2); etc.

I'll let you do the rest. Then add up all the individual variations to get the total.

 

[Eren10]

It is straightfoward: at x=0, the function jumps from 1 to 0 (var = 1); from 0 to pi/2, it goes from 0 up to 1 (var = 1); from pi/2 to 3pi/2, it goes down from 1 to -1 (var = 2); etc.

I'll let you do the rest. Then add up all the individual variations to get the total.

Thank you for your reply.

I had only used for the sin(pi*x) the function of the total variation( given in the picture, attached), because it is differentiable, for the other jumps I have used the same idea like you.

Do you certainly know that I should take max, min points of the sinus function ?

 

[mathman]

Your picture doesn't appear on click.

In any case, variations are always the absolute value of the change between max and min points, plus jumps as needed. For the sine, these are π/2 + kπ, for any integer k.

 

[Eren10]

Again, thank you. For me it is now clear.

this picture makes it also very clear, from wikipedia, As the green ball travels on the graph of the given function, the length of the path traveled by that ball's projection on the y-axis, shown as a red ball, is the total variation of the function.