Question

Question #055ec

Answer 1

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?