Answer to Question #170315 in Real Analysis for Prathibha Rose

Question

Let f be of bounded variation on [a,b] and define V is,

V(x)= {0 if x=a

V₁(a,x) if a less than or equal to x.less than or equal to b

Then show that a point of continuity of f is also a point of continuity of V and conversely

Answer 1

The total variation of α on the interval [a, b]:

$V_f(a,b)=sup\{\displaystyle\sum^n_{i=1}|f(x_i)-f(x_{i-1})|$

where $sup$ is taken over all possible partitions of [a, b].

If $x$ is a point of continuity of f, then there exists

$V_f(a,x)=sup\{\displaystyle\sum^n_{i=1}|f(x_i)-f(x_{i-1})|$

where $f(x_n)=f(x)$. (Example: $V_f(a,x)=f(x)-f(a)$ )

That is, $V_f(a,x)$ is continious at point $x$ .

If $x$ is a point of continuity of $V_f(a,x)$ , then $f(x_n)=f(x)$ has definite value, so $f(x)$ is continious at point $x$ .