Quantum mechanics is generally considered to be a theory that describes discrete behavior at the microscopic level. It introduces the concept of quantization, where certain properties such as energy, momentum, and angular momentum are restricted to discrete values known as quantized states. These quantized states are represented by discrete energy levels, and transitions between these levels occur in discrete steps.
However, it is important to note that the mathematical formalism of quantum mechanics allows for continuous probabilities and wave-like behavior. The wavefunction, which describes the quantum state of a system, is typically a continuous function that satisfies the Schrödinger equation. The probabilistic interpretation of the wavefunction, given by the Born rule, allows for the calculation of probabilities for different outcomes.
So while quantum mechanics incorporates both discrete and continuous aspects, its fundamental nature is often described as discrete due to the quantization of certain properties and the existence of discrete energy levels.