Let me offer an alternative motivation. Suppose that the universe is ultimately discrete at its smallest scales, bounded by some finite length, and everything advances in discrete 'ticks' of time. There would be no physical realisation of the real numbers.

But they would still be _useful_ as a computational device. Calculus seems intimidating and hard at first glance, but it's often easier than working with the discrete problem. If you've ever taken a probability course you may have encountered this: the binomial distribution for a large number of trials can be approximate by a Gaussian (at least, if I'm not looking too far into its tails). At a more advanced level, quants model the price process of securities as stochastic differential equations. Of course, this is nonsense - prices are in discrete number of pennies, for instance. But the error of this approximation is tolerable to produce an analytic result (the Black-Scholes formula), just so long as one remembers it isn't an exact result.

If you want a more physical example, consider the fact that we treat fluids as continua, modelled by the Navier-Stokes equations. Of course they actually consist of discrete particles, but so long as our scales of interest are greater than the mean-free-path-length, a continuous approximation is far more practical for calculations than modelling a huge number of discrete particles.