Is the composition of two functions of bounded variation again of bounded variation?
Is the composition of two functions of bounded variation again of bounded variation?
The Correct Answer and Explanation is:
Answer:
In general, the composition of two functions of bounded variation is not necessarily of bounded variation.
Explanation
To understand why the composition of two functions of bounded variation (BV) is not necessarily of bounded variation, we need to explore the concept more deeply.
A function of bounded variation on an interval [a,b][a, b] is a real-valued function ff for which the total variation Vab(f)=sup∑i=1n∣f(xi)−f(xi−1)∣V_a^b(f) = \sup \sum_{i=1}^n |f(x_i) – f(x_{i-1})|
over all partitions a=x0<x1<⋯<xn=ba = x_0 < x_1 < \cdots < x_n = b, is finite. Such functions can have jumps but cannot oscillate wildly (like the sine function with increasing frequency).
Let ff and gg be two functions of bounded variation on [a,b][a,b]. The natural question is whether the composition f∘gf \circ g is also of bounded variation.
The answer depends on additional properties of these functions. If ff is Lipschitz continuous or even just continuous, then f∘gf \circ g will typically be of bounded variation provided that gg is of bounded variation. This is because a continuous transformation of a BV function maintains control over the total variation.
However, in the general case, if both ff and gg are of bounded variation but not necessarily continuous, the composition f∘gf \circ g may not be of bounded variation. The intuition is that if gg has many jumps and ff reacts strongly (say, with large oscillations) to those jump values, then f(g(x))f(g(x)) can accumulate an unbounded total variation.
Counterexample (Sketch):
Let g(x)g(x) be a step function with many small jumps in [0,1][0,1], and let ff be defined so that it spikes sharply at each of the jump values of gg. Both ff and gg can be individually of bounded variation, but their composition accumulates large variations from the repeated spikes, making f∘gf \circ g not of bounded variation.
Conclusion:
The composition of two functions of bounded variation is not guaranteed to be of bounded variation unless additional conditions (like continuity of ff) are met.
