real analysis - Is this discontinuous function Riemann Integrable?

I have been asked to explain whether the following is Riemann integrable:

---

$$ f(x):=\begin{cases}
x & x< \frac{1}{4} \\
\frac{1}{x} & x\geq \frac{1}{4}
\end{cases}
$$
Over the interval $[-1,1]$. I was thinking that I can say this function is bound between $0$ & $4$, and I know that all bounded functions are Riemann Integrable. However, I wasn't sure if this would be a sufficient explanation (If I am even correct).

Answer

Try writing
$$
\int_{-1}^1f(x)\;\mathrm dx=\int_{-1}^{1/4}x\;\mathrm dx+\int_{1/4}^1\frac{1}{x}\;\mathrm dx

$$
Both pieces we know are integrable, indeed they are bounded and continuous.

It would probably be good to note that the integral doesn't care about discontinuities at a (or countably many) point(s).

Edit: A little more justification for why we don't need to care about the contribution at $\frac{1}{4}$, note that
$$
\left|\int_{1/4-\epsilon/8}^{1/4+\epsilon/8}f(x)\;\mathrm dx\right |\leq\epsilon/4\sup_{-1\leq x\leq 1}|f(x)|=\epsilon\to 0
$$
as we shrink $\epsilon \to 0$.