Prove the following function is Riemann Integrable. $f(x) = 1, x < \frac{1}{2}$ and $f(x) = 2, x \geq \frac{1}{2}.$

Question

Show that the function $f(x) = \begin{cases} 1 & x<\frac{1}{2} \\ 2 & x \geq \frac{1}{2} \end{cases} $ is integrable on $[0, 1]$.

I know I can break the integral up from $$ \int_0^1{f(x)dx} = \int_0^\frac{1}{2}{f(x)dx} + \int_\frac{1}{2}^1{f(x)dx} $$ and then show that the two summands is Riemann Integrable since $f(x)$ is a constant function over the two intervals.

Can I prove this without breaking up the interval. Also using the method of tagged partitions who's norm approach zero, rather than Upper and Lower Sums.