CALCULUS Understanding Its Concepts and Methods
Continuous
A function f, defined on a set S of real numbers, is continuous at a point c of S if
(intuitive) the graph of
f
doesn't jump at
c.
(in words) we can make
f(x)
as close as we want to
f(c)
simply by choosing
x
in
S
close enough to
c.
(limits) lim_{x→c }f(x) = f(c)
(symbolic) for each ε > 0 there is a δ such that, for each x in S , if |x - c| < δ, then |f(x) - f(c)| < ε.
The function f is continuous on S if it is continuous at each point of S.
A function f defined on a set S of points (x, y) is continuous at a point (c, d) of S if
In other words, we can make | f(x, y) - f(c, d) | small by making the distance
sufficiently small.
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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.