Translation Invariance of Integral

**Theorem. **Let $\displaystyle f $ be a real-valued function that is Riemann integrable on $\displaystyle [a,b] $. For a fixed $\displaystyle k \in \mathbb{R} $, let $\displaystyle g_k $ be the function $\displaystyle g_{k}(x) = f(x+k) $ where the domain of $\displaystyle g $ is appropriately chosen so that $\displaystyle x $ is in the domain of $\displaystyle g_k $ if and only if $\displaystyle x+k $ is in the domain of $\displaystyle f $. Then $\displaystyle g_k $ is Riemann integrable on $\displaystyle [a-k, b-k] $ and $\displaystyle \smallint_{a-k}^{b-k} g_k = \smallint_{a}^{b} f $.

**Proof**. Let $\displaystyle \epsilon >0 $. Choose $\displaystyle \delta >0 $ such that $\displaystyle \delta < \frac{\epsilon}{2} $. So basically we consider $\displaystyle \mathcal{R}(g_k, P) = \sum_{i=1}^{n} g_{k}(x_{i}^{*})(x_{i}-x_{i-1}) $? And then show that $\displaystyle |\mathcal{R}(g_k,P)-\mathcal{R}(f,P)| < \epsilon $?