To analyze this question, let’s go through each option, focusing on the properties of the function $f(x)$ defined as:

[ f(x) = \begin{cases} x - \frac{1}{2} & \text{if } x \in \mathbb{Q} \cap [0,1], \ 2x & \text{if } x \notin \mathbb{Q} \cap [0,1]. \end{cases} ]

Step-by-Step Analysis of Each Option

1. Continuity of $f(x)$:

   • For a function to be continuous, the values must not "jump" between rational and irrational numbers.
   • Here, $f(x)$ takes different values depending on whether $x$ is rational or irrational in $[0,1]$. The rational points have the form $x - \frac{1}{2}$, while irrational points have $2x$.
   • This setup makes $f(x)$ discontinuous on $[0,1]$ because rational and irrational points are dense in this interval, causing $f(x)$ to “jump” at every point.

   Thus, option (d) is correct: $f$ is not continuous.

2. Riemann Integrability of $f(x)$:

   • A function is Riemann integrable on a closed interval if it is bounded and has a set of discontinuities of measure zero.
   • Since $f(x)$ is discontinuous everywhere on $[0,1]$ (due to the different values for rational and irrational points), it fails to meet the requirement for Riemann integrability.

   Hence, option (c) is also correct: $f$ is not Riemann integrable.

3. Evaluating the Integral $\int_0^1 f(x) \, dx$:

   • Since $f(x)$ is not Riemann integrable, we cannot compute the integral $\int_0^1 f(x) \, dx$ in the Riemann sense.
   • Thus, both options (a) and (b) are incorrect because no Riemann integral can be defined for this function on $[0,1]$.

Conclusion

The correct options are:

• (c) $f$ is not Riemann integrable
• (d) $f$ is not continuous

Would you like more details on any of these points? Here are related questions for further practice:

1. What makes a function Riemann integrable on a closed interval?
2. How does the density of rationals and irrationals in $[0,1]$ impact the continuity of functions like $f(x)$?
3. Can a function be Lebesgue integrable if it is not Riemann integrable?
4. How would the definition of $f(x)$ change if it were only defined on either rational or irrational points exclusively?
5. What are examples of functions that are discontinuous on all points of an interval but still Riemann integrable?

Tip: For Riemann integrability, focus on the set of discontinuities; if it has measure zero, the function can still be integrable.