Riemann integrability of a piecewise function over $[1, 7]$ I have the following function 
$f(x) = \begin{cases} 2 & \text{if } 1 \leq x \leq 2, \\ 3 & \text{if } 2 < x \leq 4,\\ 1 & \text{if } 4 < x \leq 7. \end{cases}$
How do I determine if the function is Riemann integrable within $[1, 7]$? 
I think I have to check if the lower and upper integrals of $f(x)$ match but I’m unsure how to approach.  

Since $\int_a^b f(x) dx = \int_a^cf(x) dx + \int_c^b f(x)dx$ you can split this integral into three parts, as $\int_1^7 f(x) dx = \int_1^2 f(x) dx + \int_2^4 f(x)dx + \int_4^7 f(x) dx$. Since $f(x)$ is constant on each of these intervals they are easily integrated.
