How do I show that if $f$ is bounded and integrable on $\mathbb{R}$, then $g(t) = \int_t^{t+1} f(x) dx$ is continuous? Usually functions of the form of $g(t)$ tell me I should use the fundamental theorem of calculus, but I don't think that applies here because I'm not given that $f$ is continuous.
I know from the definition of continuity that I need to show for some $\epsilon$ and some $x \in \mathbb{R}$, there exists a $\delta$ such that if $|x - y| < \delta$, then $|g(x) - g(y)| < \epsilon$.
$f$ is bounded, so there is some $M$ so that $f(t) < M$ for all real numbers $t$. Also, since $f$ is integrable, I can find two sequences of partitions $P_{x,k}$ and $P_{y,k}$ so that 


*

*$\lim_{k \to \infty} |P_{x,k}| = 0$ and $\lim_{k \to \infty} |P_{y,k}| = 0$, and

*$\int_x^{x+1} f = \lim_{k \to \infty} U(f, P_{x,k})$ and $\int_y^{y+1} f = \lim_{k \to \infty} U(f, P_{y,k})$.
For any $k \in \mathbb{N}$, the sequences $P_{x,k}$ and $P_{y,k}$ may have different numbers of points, so I used $N_x$ and $N_y$ for these numbers of points, for some arbitrary $k$.
Ploughing onwards, I tried to work out what my $\delta$ should be be working backwards from the definition of continuity, but that gave me this:
\begin{align}
|g(x) - g(y)|
&= \left| \int_x^{x+1} f(u)du - \int_y^{y+1} f(u)du \right| \\
&= \left| \lim_{k \to \infty} (U(f, P_{x,k} - L(f, P_{x,k}) - \lim_{k \to \infty} (U(f, P_{y,k} - L(f, P_{y,k}) \right| \\
&= \left| \lim_{k \to \infty} \sum_{i=0}^{N_x - 1} (M_{x,i} - m_{x,i})(x_{i+1} - x_i) - \lim_{k \to \infty} \sum_{i=0}^{N_y - 1} (M_{y,i} - m_{y,i})(y_{i+1} - y_o) \right| \\
&< \left| 2M \lim_{k \to \infty} \sum_{i=0}^{N_x - 1} (x_{i+1} - x_i) - 2M \lim_{k \to \infty} \sum_{i=0}^{N_y - 1} (y_{i+1} - y_i) \right| \\
&= \left| 2M \lim_{k \to \infty} (x_{N_x} - x_0) - 2M \lim_{k \to \infty}  (y_{N_y - 1} - y_0) \right| \\
&= \left| 2M \lim_{k \to \infty} (1) - 2M \lim_{k \to \infty}  (1) \right| \\
&= 0
\end{align}
Sadly this didn't give me a $\delta$, and I'm not sure where I used the assumption that $|x - y| < \delta$, whatever that $\delta$ might be. 
Did I take a wrong turn somewhere?

I think you're making it more difficult than it has to be. We have $$|g(x)-g(y)|=\left|\int_x^{x+1}f(t)dt -\int_y^{y+1}f(t)dt\right|\\=\left|\int_x^yf(t)dt-\int_{x+1}^{y+1}f(t)dt\right|$$ Now use that $f$ is bounded.
