Quantum mechanics shows us that nature is discontinuous. An electron can exist around an atom only in certain places and the space between these places is forever empty. In space electrons pop in and out of existence from seemingly nowhere.
Natch, that would be the Bohr model, which in fact is experimentally invalid. In fact, when you use the Schroedinger equation to analyze the hydrogen atom (or any equivalent hydrogen-like model of the atom), there are an infinite number of possible states which have no nodes - i.e. the electron can truly be anywhere in the universe (although with a vanishingly small probability outside the "confines" of the atom), in continuous fashion.
By the way (and this is one of my massive gripes with non-mathematicians who try to use mathematical language without giving squat about what it actually
means) "discontinuous" is not equal to "nonlinear". Plenty of nonlinear functions are continuous.
Mathematical excursion:
Linearity is roughly "additivity": loosely speaking, a linear function (or functional, or space, or whatever) is one for which it doesn't make a difference whether you add things together before you throw them into the mathematical machinery, or after. The temperature scale between inches and centimeters is a good example of a linear scale. Suppose I have a 3-inch-long stick and an 8-inch-long stick (curses on you Americans!). I add them together, and then convert to centimeters: an 11-inch-long stick is 28cm. Now let's convert to centimeters, and then add them together: 3 inches is 7.7cm, and 8 inches is 20.3cm, and so the total length is 28cm. It doesn't make a difference whether I add before converting or after: the conversion is linear.
In contrast, a good example of a nonlinear scale is the decibel system. A typical person talks at a volume of 50dB; a jackhammer blasting 1m away drills at 100dB. Egads! Does that mean that two people talking at me at the same time have the same loudness as a jackhammer? (Joke punchline: Depends on the person.) In reality, the decibel scale is a
logarithmic scale, and thus 50dB worth of sound + 50dB worth of sound is not 100dB worth of sound. It makes a difference whether I add sound together before measuring their decibel strength, or whether I measure two sounds' decibel strengths and then "add" them together.
Continuity and discontinuity, by contrast, have to do with whether you can draw a "smooth graph" of a function. Here's a decent example: let's say I had $100 in my bank account, and make a deposit of $100 more. My account balance is
discontinuous: it was $100 yesterday, and now it is $200, but you can't find a time when it was, say, $150, or $175, or $120: it's either $100 or $200. (Unfortunately.) But suppose I'm walking away from the ATM when a villain from Acme Inc. suddenly drops a safe from 10m up onto my head. Someone videotapes the whole incident and puts it up on Youtube: so a bored Youtube viewer can find a time when the safe was 5m above me, or 2m, or 3cm, or (in classical theory, and assuming Youtube's resolution increases dramatically between now and then) 1 micrometer above my head: physical quantities are normally
continuous.
By now it should be apparent that discontinuity and non-linearity don't go together. For example, let's say I buy a computer which is interest-free until 2010, and with compounding interest after that. My balance on it will be
discontinuous and
linear until 2010, and
discontinuous and
nonlinear after that. Meanwhile, both the conversion scales I mentioned above are
continuous (you can apply them to arbitrarily small gradations), but the length scale is
linear while the sound scale is
nonlinear.
-excursion over
Basically, check your math?
