By Kenneth S. Miller.
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Additional resources for An introduction to the calculus of finite differences and difference equations
X/ is also discontinuous at x D a. x/ D x C x is continuous at x D a. For example, if a D 2: 4 2 -6 -4 -2 0 0 2 4 6 2 4 6 -2 -4 4 2 -6 -4 -2 0 0 -2 -4 ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2012/7/28 — 0:02 — page 47 — #57 ✐ ✐ 47 3. x/ is also discontinuous at x D a. x/ D sin2 x ; x2 if x ¤ 0 1; if x D 0 is continuous at the point x D 0. 4 A function always has a local maximum between any two local minima. Counterexample The functions yD x 4 C 0:1 x2 and y D sec2 x have no maximum between two local minima: ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2012/7/28 — 0:02 — page 49 — #59 ✐ ✐ 49 3.
A/ is a point of inflection for the graph of the function. a/ is a point of inflection on the function’s graph, then the second derivative is zero at that point. x/ D 1. x/ also exists. x/ also exists. x/ is differentiable at the point x D a, then its derivative is continuous at x D a. x/ is positive at the point x D a, then there exists a neighborhood about x D a where the function is increasing. a; b/, then in a sufficiently small neighborhood of the point x D c the function is increasing for all x < c and decreasing for all x > c.
C/ D 0. c/ D N . c/ D N .
An introduction to the calculus of finite differences and difference equations by Kenneth S. Miller.