Question #055ec

1 Answer
Sep 21, 2017

The integral may converge or diverge.

Explanation:

Example:
Let f(x) = x
Let a = 0
Let b = -1.
For this function and these values of a and b...
f(a + x) = f(x) = x and f(b + x) = f(x - 1) = x - 1.
Therefore
f(a + x) - f(b + x) = 1 for all x, and we have
int_-oo^oo(f(a + x) - f(b + x))dx = int_-oo^oo1dx, which diverges.

However if:
Let f(x) = 1
Let a = 0
Let b = -1.
For this function and these values of a and b...
f(a + x) = f(x) = 1 and f(b + x) = f(x - 1) = 1.
Therefore
f(a + x) - f(b + x) = 0 for all x, and we have
int_-oo^oo(f(a + x) - f(b + x))dx = int_-oo^oo0dx = 0.

Were there limitations on f, or am I understanding your question correctly?