02.07.08
Math 1a roundup for February 6
Continuity is the property that the limit of a function near a point is the value of the function near that point. It’s one of many “nice” properties a function can have. Functions can fail to be continuous in a number of different ways, including jumping from one value to another, having a simple “removable” discontinuity, or blowing up to infinity near a point.
Lots of times we want to model a situation with a function, and we assume the function is continuous to make our lives easier. For instance, the population of a bacteria culture in a dish. Clearly there are only whole numbers of organisms in the culture, but we pretend the population function of time is continuous because relative to the population size, an error of one by rounding isn’t that much.
An important consequence of continuity is the intermediate value theorem, which says that a continuous function on an interval assumes all the values in between the values at the endpoints. This theorem has important consequences, for instance:
- the square root of two exists
- at some point in your life your height in inches equaled your weight in pounds
- A table on an uneven floor can always be turned so that it won’t wobble
- Right now there are points on opposite sides of the world with the same temperature!
Blogged with Flock
Tags: math, math1a, function, continuity, limit