Let $f :[0,1] \rightarrow \mathbb{R}$ and $g :[0,1] \rightarrow \mathbb{R}$ be defined as follows:
$f(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases}$
$g(x) = \begin{cases} 0 & \text{if } x \text{ is rational} \\ 1 & \text{if } x \text{ is irrational} \end{cases}$
Then:

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