I often like to fool around on graphing calculators when I am bored. A function that can be very amusing is
$f(x) = rand \times \sin x$
Now, on my TI-84 Plus, this looks obviously discontinuous because of the poor resolution. And there can be spikes when the value of $rand$ is higher or lower than average, however I feel as the scale of the window $\to \infty $ that the probability of the function being discontinuous would decrease because the effects on $sin x$ would be less dramatic. Now I understand that at $x = \frac{\pi}{2}$, $f(x)$ may equal $1$, and then at $\lim_{x \to \frac{\pi}{2}^+}f(x)$ could be equal to $0$ if $rand = 0$, however there is a point inbetween $\frac{\pi}{2}$ and $\lim_{\frac{\pi}{2}^+}$, and this can be reiterated an infinite amount of times. Does this make $f(x)$ continuous?
$f$ as you originally wrote is not a function in the mathematical sense because given a fixed $x$, $f(x)$ does not have a fixed value. Repeated evaluations would yield different results.
The way to formalize this is to properly introduce a random variable, say $Y$, then make $f$ a function of two variables: $f(x, y) = y \sin x$. The function $x \mapsto f(x, Y)$ is the proper definition of your original function. Note that you need to specify the distribution of $Y$ that corresponds to your $rand$ function. For example, if $rand$ is a continuous uniform random variable between $0$ and $1$, then the probability density function of $Y$ is $y \mapsto 1$ for $y \in [0, 1]$.
$f$ is now a function in the conventional sense. You can talk about its values, limits, partial derivatives, etc. We can also put randomness into our statements like $\Pr(f(\pi/2, Y) \le 1/2) = 1/2$, or $E\big[\frac{\partial f}{\partial x}(x, Y)\big] = \frac 12\cos x$.