Unformatted text preview: A function f is diﬀerentiable at point a (excluding endpoints and isolated points in the domain of f ) if f ± ( a ) exists. DEFINITION 2.21 Diﬀerentiable A function f is diﬀerentiable if is diﬀerentiable at every point (excluding endpoints and isolated points in the domain of f ) in the domain of f . Take note that, for technical reasons not discussed here, both of these deﬁnitions exclude endpoints and isolated points in the domain from consideration. We now have a collection of adjectives to describe the very rich and complex set of objects known as functions. We close with a useful theorem about continuous functions: THEOREM 2.22 Intermediate Value Theorem If f is continuous on the interval [ a, b ] and d is between f ( a ) and f ( b ), then there is a number c in [ a, b ] such that f ( c ) = d . This is most frequently used when d = 0. EXAMPLE 2.23 Explain why the function f = x 3 + 3 x 2 + x2 has a root between 0 and 1....
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 Fall '07
 JonathanRogawski
 Math, Calculus, Continuity, Derivative, Rate Of Change, Continuous function

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