Answer to Question #170333 in Real Analysis for Prathibha Rose

Question #170333

If f:[a,b] to R is monotonic ,prove that f is of bounded variation on [a,b]?


1
Expert's answer
2021-03-22T08:39:21-0400

Solution


A function"f : [a,b] \u2192R" is monotonic if and only if "f" is a function with bounded variation and "\\overset{b}{ \\underset{a}{V}} = | f(b) \u2013 f(a) |."


A function"f : [a,b] \u2192R" is called with bounded variation on "[a,b]" if there is "M > 0" such that for any partition "\u2206 = (a = x_0 < x_1 < ... < x_n = b)" of the interval [a,b] we have:

"\\overset{b}{ \\underset{a}{V}} = sup \\{V_\u2206(f) |\u2206 \\ division \\ of\\ [a,b]\\}"

is called the total variation of the function f on the interval "[a,b]" .


Given a monotonic function "f: [a,b]\u2192R" and a partition "P=\\{a=x_0< x_1< x_2< ... < x_n< b=x_n+1\\}"

of "[a,b]", the variation of "f" over "P" is;


"V_P(f) =\u2211_{j=0}^n|f(x_j)\u2212f(x_{j+1})|."

"f" is of bounded variation if the numbers "V_P(f)" form a bounded set, as "P" ranges over the set of all partitions of "[a,b]". We denote the supremum of the "V_P(f)" over all partitions "P" by "V_{ab}(f)", the variation of "f" from "a" to "b".


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