Appendix F
On Continuous Functions
What Functions are Continuous
The following theorem will help you quickly identify
continuous functions in any calculus course that you
take.
Therem F.1.[Identification of Continuous Functions]
- If f(x) is differentiable at x = x1
then f(x) is continuous at x = x1.
In other words, if f¢(x1)
is defined at x = x1 then f(x)
is continuous at x = x1.
- If f(x) and g(x) are
differentiable at x = x1, and c
is a real number, then c·f(x), f(x)+g(x),
f(x)-g(x),
f(x)·g(x) are also
differentiable (and hence continuous) at x = x1.
- If f(x) and g(x) are
differentiable at x = x1, and g(x1)
¹ 0, then [1/(g(x))]
and [(f(x))/(g(x))], f(x)·g(x)
are also differentiable (and hence continuous) at x
= x1.
- If f(x) and g(x) are
continuous at x = x1, and c
is a real number, then c·f(x), f(x)+g(x),
f(x)-g(x),
f(x)·g(x) are also
continuous) at x = x1.
- If f(x) and g(x) are
continuous at x = x1, and g(x1)
¹ 0, then [1/(g(x))]
and [(f(x))/(g(x))], f(x)·g(x)
are also continuous at x = x1.
The assertions in this theorem are consequences of the
previous theorem on the algebraic properties of limits. This
theorem indicates how arithmetic operations on continuous or
differentiable functions respectively yield further
continuous or further differentiable functions. First
books on calculus often contain a section or two explaining
why the assertions in this theorem hold. So the proofs
are omitted. The aim of this work is to complement other
texts. | |
the Real Analysis appendices of
Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
Presenting Appendices from Volume 3, Why
Slopes and More Math, If the epsilon-delta viewpoint of
limits, continuity and convergence is not yet comfortable, see Chapters 14
to 19 in Volume 3 are related.
A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits, Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II
Proofs of one-sided theorems could be of interest in
the study of 2D topology.
Vol 1A Logic Postscripts
online only include
Proof
by Absurdity alias proof by contradiction
How
the demand for consistency supports the law of the excluded middle
Reality
versus or with the aid of Imagination
|