The function $f(x)$ satisfies
\[f(xy) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(0) \neq 0.$  Find $f(10).$
I notice that this is a functional equation, which means that I need to find a general formula for $f(x)$ that satisfies the given equation for all $x$ and $y.$ One strategy for solving functional equations is to plug in some special values of $x$ and $y$ and see what happens. For example, if I plug in $x = 1,$ I get $f(1y) = f(1) f(y),$ which simplifies to $f(y) = f(1) f(y).$ This means that either $f(1) = 1$ or $f(y) = 0$ for all $y.$ But the problem says that $f(0) \neq 0,$ so I can rule out the second option. Therefore, $f(1) = 1.$ This might be useful later, so I'll keep it in mind. Another special value to try is $x = 0.$ Plugging in $x = 0,$ I get $f(0y) = f(0) f(y),$ which simplifies to $f(0) = f(0) f(y).$ This means that either $f(0) = 0$ or $f(y) = 1$ for all $y.$ But again, the problem says that $f(0) \neq 0,$ so I can rule out the first option. Therefore, $f(y) = 1$ for all $y.$ This means that $f(x)$ is a constant function that always equals 1, regardless of the value of $x.$ To check, I can plug this into the original equation and see that it works: $f(xy) = f(x) f(y) = 1 \cdot 1 = 1$ for all $x$ and $y.$ So I have found the function $f(x),$ and I can use it to find $f(10).$ By the same logic as before, $f(10) = 1.$ 