Can the boolean difference of boolean function be 0?

Question

Suppose we have a boolean function $$f(x_1, x_2, ..., x_n)$$
The boolean differential of this function is defined as follows
$$ \frac{df(x_1, x_2, ..., x_n)}{dx_i} = f\big|_{x_i = 0 } \oplus f\big|_{x_i = 1 } = \\ f(x_1, x_2, ..., x_{i-1}, 0, x_{i+1}, ..., x_n) \ \oplus \ f(x_1, x_2, ..., x_{i-1}, 1, x_{i+1}, ..., x_n) $$
Can this differential be equal to 0 and what does it mean when it is?

Answer 1

Hint: if $X$ and $Y$ are truth values, then $X \oplus Y = 0$ means that $X$ and $Y$ are the same (check the truth table, if you didn't know that). Can you now see the answer?

The boolean differential $df/dx_{i}$ is (the constant function) $0$ iff the value of $f$ is independent of $x_i$.