The translation property (Theorem 1.7) states

Prove that the following is equivalent to the translation property:

* Proof. * Let and . Then, by the translation property we have:

using in the theorem. Substituting and back into the equation, we obtain,

What if the function f isn’t defined in the interval [a+c, b+c]?

I think there’s a typo on the first integral it should be dx not dt, i think.