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Unformatted text preview: be continuous where it has a ‘break’ or ‘hole’
in the graph; everywhere else, the function is continuous. The function is continuous at the ‘corner’
and the ‘cusp’, but we consider these ‘sharp turns’, so these are places where the function fails
to be smooth. Apart from these four places, the function is smooth and continuous. Polynomial
functions are smooth and continuous everywhere, as exhibited in graph on the right. ‘hole’
Pathologies not found on graphs of polynomials
The graph of a polynomial The notion of smoothness is what tells us graphically that, for example, f (x) = |x|, whose graph
is the characteric ‘∨’ shape, cannot be a polynomial. The notion of continuity is what allowed us
to construct the sign diagram for quadratic inequalities as we did in Section 2.4. This last result is
formalized in the following theorem.
Theorem 3.1. The Intermediate Value Theorem (Polynomial Zero Version): If f is a
polynomial where f (a) and f (b) have diﬀerent signs, then f has at least one zero between x = a
and x = b...
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