Question

Show that a function of bounded variation on [a,b] is bounded therein

Answer 1

 Let f be a function of bounded variation on the interval [a,b].

Then there exists a positive real number M>0 such that for all partitions P∈ P[a,b]

 we have that:

$$V_f(P)=\Sigma_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$$

For all x∈[a,b] consider the partition P={a,x,b} (where P={a,b} if x=a or x=b).

Then:

$$V_f(\{a,x,b\})=|f(b)-f(x)|+|f(x)-f(a)|\leq M$$

Hence we have that $|f(x)-f(a)|\leq M,$ so for all x∈[a,b] we have that $|f(x)|\leq |f(a)|+M$ ,

So, f is bounded on [a,b]