Definition:Bounded variation and if f is monotonic on [a,b] then f is Bounded variation on [a,b]

[00:00:00]hello viewers I discussed today this idea of bounded variations on an interval first we define the function on the interval specific most interval and then we take the partitions on it and segment wise we partitioned these functions defined on this all interval a B and this interval particularly if we consider like this a and B and we take the several partitions like this and if we are taking like the point if we are taken on this point like this x equals to this point is equals to X 0 if we say then suddenly it will be X 1 another is X 2 in this manner we are finally getting the point of xn in this manner and correspondingly we are getting this result of F of xn and in this way we find some pictorial result and the representations all these type of specific representations suddenly then the arbitrary length is defined actually all these type of a result and it will occur whatsoever the cobby's like this this point here it is f of x0 and if we consider it as f of x1 in this manner here it is f of x2 and here it is a finally F of xn in this way now let's construct our definition spot for the part of definition of bounded variation net where specifically if be defined on this particular interval that is a closed interval a B and if we take this partition like P and all this point x over x0 x1 to xn 0 X 1 up to xn is the partition and this partition of a B and if we write it writing it as Delta F of kaykai's arbitrarily define the variable result K is one two three four we are getting this variation F of XK minus f of X K minus one and getting this result here for K is equal to 1 to up to him and if there exist a positive positive number say this him such that as a bounded field we are taking summation K is equal to 1 2 up to n mod of Delta F K that will be less than equals to n for all partition of a B then we can say a is said to be of bounded variation or theory whatever may be the swab you have deviated bounded variation in this interval a B so in the way we construct our definitions of bounded variations the function is defined on this particular interval a B pictorially at first we represent it this partition all this of this result and this partition suddenly we have taken this interval a delta of FK there's a change of this interval there's a functional value changes this is called the variations and we are changing in these variations and taking a constant there is a bounded figure we are getting this result this is ultimate definitions of bounded variations now let us consider interesting result very very interesting material it has a theorem part may be a considered as a theorem if if is monotonic and this monotonic on a B then if is of bounded variation or BV on a B then how to prove this result at first if we are taking the part of this let a be increasing then for every partition for every partition of a B we have then of FK that's a non negative and hence summation K is equal to 1 to up to n mod of Delta FK and that will be summation K is equal to 1 to up to n Delta FK if we are breaking this value for K equals 1 to up to n that is f of XK minus F of XK minus 1 and suddenly if we are breaking this result of f of X 1 minus f of X 0 in this way plus f of X 2 minus f of X 1 plus of 2 if of it is f of X n minus f of X n minus 1 and suddenly all a canceled finally we find f of xn minus f of X 0 and this result will be suddenly xn point will be a B minus F a we are getting this result which implies summation K is 1 2 up to n mod of F of K and suddenly that will be less than equals to M where we can take a B minus F is bounded function that is less than equals T hence the proof so we end our discussions here if you like this video like it share it and always subscribe the channel thank you