Question

Let \( f: [0, 1] \to \mathbb{R} \) and \( g: [0, 1] \to \mathbb{R} \) be defined as follows:
 
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \cap [0, 1] \\ 0 & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cap [0, 1] \end{cases} \]
The function \( g(x) \) is defined as:
\[ g(x) = \begin{cases} 0 & \text{if } x \in \mathbb{Q} \cap [0, 1] \\ 1 & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cap [0, 1] \end{cases} \]
Then:

• A. \( f \) and \( g \) are continuous at \( x = \frac{1}{2} \).
• B. \( f + g \) is continuous at \( x = \frac{2}{3} \) but \( f \) and \( g \) are discontinuous at \( x = \frac{2}{3} \).
• C. \( f(x) \cdot g(x)>0 \) for some \( x \in (0, 1) \).
• D. \( f + g \) is not differentiable at \( x = \frac{3}{4} \).

Hint:

Be careful with functions defined piecewise on rationals and irrationals. Their continuity properties are often counter-intuitive.