Yes and no. Clearly, stock prices (or prices of any asset) are not observed continuously. This applies to both, the value (price) dimension and the time dimension.

This however does not mean that we can't model stock prices as a time and space continuous process. Frequently, time and space continuous approaches are more elegant and yield nicer results. Furthermore, continuous models are often limits of the discrete models. Anyway, the implementation of continuous models requires you to discretise the model (because you only have discrete data sets and computers can only work with discrete sets).

Note that we can, by the way, record prices with higher precision than cents or pence. Look at currencies which can be traded with a finer grid and we could (technically) use arbitrary fine partitions of the positive real axis. But yes, you can never observe a stock trading at \$$\pi$. But models are simply easier and nicer if you allow for a continuous range.

Whether you see stock prices as a continuous process which we merely record discretely or whether you believe stock prices are discrete objects which we simply model continuously is almost a philosophical question.

Here some examples:

  • Discrete time, discrete state space

Simple Random Walk

  • Discrete time, continuous state space

Gaussian Random Walk

  • Continuous time, discrete state space

Poisson Conting Process

  • Continuous time, continuous state space

Brownian motion