Piecewise functions: Got an example of a real world piecewise function? Looking for something beyond a contrived textbook problem concerning jelly beans or equations that do not represent anything concrete. Not just a piecewise function for its own sake.  Anyone?
 A: How about a monthly mobile phone plan? Your cost is constant (maybe $50/month) as long as you stay under the allotted minutes (maybe 1000/month), but then you are charged per minute after.
A: Shock waves. Properties of materials before and after phase transitions. 
A: Any disturbed physical system. If you're dealing with circuits you'll often want to solve an equation that involves switches. E.g, letting a capacitor charge for 1 ms and then switching the connection to a closed loop where it discharges. That switch naturally gives a piecewise-defined function. Similar ideas go for physical systems involving collisions, such as a bouncing ball. They come up all the time.
A: Depth of a swimming pool as function of the distance from one end:

A: As Wim mentions in the comments, piecewise polynomials are used a fair bit in applications. In designing profiles and shapes for cars, airplanes, and other such devices, one usually uses pieces of Bézier or B-spline curves (or surfaces) during the modeling process, for subsequent machining. In fact, the continuity/smoothness conditions for such curves (usually continuity up to the second derivative) are important here, since during machining, an abrupt change in the curvature can cause the material for the modeling, the mill, or both, to crack (remembering that velocity and acceleration are derivatives of position with respect to time might help to understand why you want smooth curves during machining).
A: I always thought the horizontal distance from a wall to an object is a good example of a piecewise continuous function (of the height):

Here the various red lines show the various distances and the green line marks the point of discontinuity.
A: Computing income taxes in a bracketed system.
A: Piecewise constant functions come up all the time in the design and analysis of digital circuits (see square waves, for example).  The finite element method is a very widely used technique that approximates solutions of differential equations as piecewise linear functions.
A: Postal rates (as a function of weight, in ounces):

Look at the price of a $1.9$ oz letter, then $1.99$ oz letter, then $2.01$ oz letter, etc.
A: The free-fall acceleration on and in the earth must be modeled piece-wise, even in the simplest model of constant density.

A: Cab fares are a piecewise constant function of distance travelled. 
A: The classic example is friction---say for a block sitting on a rough horizontal plane and subject to a continuously increasing horizontal force. The frictional force rises steadily to match the applied force, and then it drops back a bit to a constant value when the block begins to slide.
A: Something like "buy $5$, and get each one after that at half price" say at a grocery store or clothing store. For example, let $C(x)$ be the cost of the item for the consumer and $p(x)$ be the price (assume it is constant.)
Then, 
$C(x) = 5p |x\le5$ or $C(x) = 5p + (x-5)\cdot\frac{1}{2}p|x>5$
A: Body must be at least 30 characters.

A: What about the travel time to a fixed destination in a scheduled public transport system? It is a piecewise linear function of the departure time with slope $-1$.
A: Bussiness is a good example. The increase in salary shows a piecewise function.
A: I liked the rates in a parking garage as a step function or there is the wages at a job - when you include overtime you have a piecewise function of your pay as a function of the hours you worked.
A: You want to visit two shops in the High Street. You drive your car to the High Street, park it, walk to each shop and back to your car. What distance do you have to walk as a function of where you park? 
