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share two properties which help distinguish them from other animals in the algebra zoo: they are
continuous and smooth. While these concepts are formally deﬁned using Calculus,15 informally,
graphs of continuous functions have no ‘breaks’ or ‘holes’ in their graphs, and smooth functions have
no ‘sharp turns.’ It turns out that these traits are preserved when functions are added together, so
general polynomial functions inherit these qualities. Below we ﬁnd the graph of a function which is
13 We ignore the case when n = 1, since the graph of f (x) = x is a line and doesn’t ﬁt the general pattern of
higher-degree odd polynomials.
And are, perhaps, the inspiration for the moniker ‘odd function’.
In fact, if you take Calculus, you’ll ﬁnd that smooth functions are automatically continuous, so that saying
‘polynomials are continuous and smooth’ is redundant. 3.1 Graphs of Polynomials 185 neither smooth nor continuous, and to its right we have a graph of a polynomial, for comparison.
The function whose graph appears on the left fails to...
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