lecture04 - 1 1.1 Continuity Definitions Last time I went...

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1 Continuity 1.1 Definitions Last time, I went over the direct substitution property. That says that for polynomials and rational functions, as long as a is in the domain, lim x a f ( x ) = f ( a ). It turns out that this property works for a lot of different types of functions. If a function has the dsp, we say that it is continuous . Definition 1. A function f is continuous at a number a if lim x a f ( x ) = f ( a ) There are three steps to showing continuity, 1. a is in the domain of f . This is very important. If f ( a ) isn’t defined, then we can’t plug a in to find the limit. However, the limit may still exist. 2. lim x a f ( x ) must exist. Think of piecewise functions. 3. lim x a f ( x ) = f ( a ) That’s the definition. What does continuity mean, though? A function is continuous if a small change in x causes a small change in f ( x ). That helped, I’m certain. Think of driving a car. Over a very small period of time, the car will only move a little bit. It won’t suddenly jump 10 feet. The temperature is a continuous function.
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