Question

Define bounded and unbounded variation. Show that every function which is of bounded variation is bounded

Answer 1

Bounded variation $\rightarrow$ A function is said to be bounded variation if

over the closed interval $x\in[a,b]$ , the function is finite.

$\bigstar$ A function of bounded variation, is a real-valued function 

$\bull$ whose total variation is bounded (finite)

$\bull$ the graph of a function having this property is well behaved in a precise sense.

Unbounded Variation $\rightarrow$ Unbounded variation is is just opposite to bounded variation , if a function $x\in[a,b]$ , the function is $\infin$ .

And properties are also opposite to bounded variation.

Every function which is of bounded variation is bounded

As

$\bigstar$ A function of bounded variation, is a real-valued function whose total variation is bounded (finite)

These lines states that , function which is bounded in an interval with finite , is bounded variation .

And

$\bigstar$

$\bull$ Let f be the function defined : [a, b] → R, f is of bounded variation if and only if

$\bull$the f is the difference of two increasing functions.

and

$\bull$ thus v(x) − f(x) is increasing.

$\bull$ The limits f(c + 0) and f(c − 0) exists for any c ∈ (a, b).

$\bull$ The set of points where f is discontinuous is at most countable.