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Learn about continuity in functions, where a function has no breaks, jumps, or holes. Explore the Intermediate Value Theorem and how a continuous function cannot skip over values.
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Lesson 1:____ Section 1.7 Introduction to Continuity A function is said to be continuous over an interval if it has no breaks, jumps, or holes. (the pencil never leaves the paper!) Ex. Given that function g is continuous, if we know that g(10) is positive, while g(15) is negative, what can we conclude must have occurred?
The Intermediate Value Theorem Suppose f is continuous on a closed interval [a,b]. If k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k A continuous function cannot skip over values! As we move from a to b, the function takes on every output value between f(a) and f(b).
If a function is continuous, nearby values of the independent variable give nearby values of the function. This means that tiny errors in the independent variable should lead to tiny errors in the value of the function. If f(x) is continuous at x = c, then as x approaches c, the values of f(x) will approach f(c). As we get really close to a particular input, we are also getting really close to a particular output. Hmm… this language sounds familiar…
PROOF of CONTINUITY A function is continuous at c if the following 3 conditions are met: • exists • exists f(c) c This is called the “definition of continuity”
