Real-valued Functions:Continuity & Uniform Continuity

Josh EngwerTexas Tech University

josh.engwer@ttu.edu

July 5, 2011

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The Real Line R :

• Assume function f : I → R, where I ⊆ R is an interval. Assume {xn} ∈ I is a sequence of real numbers in I .

EPSILON DEFINITIONS:

• f is continuous at point c ∈ I ⇐⇒ ∀ε > 0,∃δ > 0 s.t. |x− c| < δ ⇒ |f(x)− f(c)| < ε.

• f is uniformly continuous on I ⇐⇒ ∀ε > 0, ∃δ > 0 s.t. ∀x, y ∈ I, |x− y| < δ ⇒ |f(x)− f(y)| < ε.

EASIER DEFINITIONS:

• f is continuous at point c ∈ I ⇐⇒ ∀ {xn} ∈ I, {xn} → c⇒ {f(xn)} → f(c).

• f is continuous on I ⇐⇒ f is continous at every point of I .

• f is discontinuous at point c ∈ I ⇐⇒ ∃ {xn} ∈ I s.t. {xn} → c but {f(xn)} 6→ f(c).

MANIFESTATIONS OF UNIFORMLY CONTINUOUS FUNCTIONS:

• f is continuous on closed bounded interval I ⇒ f is uniformly continuous on I .

• f is Lipschitz on I ⇒ f is uniformly continuous on I .

• Let f be uniformly continuous on I . Then {xn} is Cauchy in I ⇒ {f(xn)} is Cauchy in R.

INTERMEDIATE VALUE THEOREM:

• Let f be continuous on I . Then k ∈ (a, b) ⊆ I satisfies f(a) < k < f(b)⇒ ∃c ∈ (a, b) s.t. f(c) = k.

PRESERVATION OF INTERVAL MAPPINGS:

• f is continuous on interval I ⇒ image f(I) is also an interval.

• f is continuous on closed bounded interval I ⇒ image f(I) is also a closed bounded interval.

COMBINATIONS OF CONTINUOUS FUNCTIONS:

• f, g are continuous on I ⇒ f + g, f − g, fg, kf, |f | are all continuous on I , where k ∈ R.

• f, g are continuous on I and g(x) 6= 0 ∀ x ∈ I ⇒ f/g is continuous on I .

MONOTONICITY AND CONTINUITY:

• f is monotone on I ⇒ set of points in I where f is discontinuous is at-most countable.

• f is monotone on I and continuous at point c ∈ I ⇐⇒ there’s no jump discontinuity at point c.

• f is strictly monotone and continuous on I ⇒ inverse f−1 is strictly monotone and continuous on f(I).

DIFFERENTIABILITY AND CONTINUITY:

• f is differentiable at point c ∈ I ⇒ f is continuous at point c.

INTEGRABILITY AND CONTINUITY:

• f is continuous on [a, b]⇒ f is Riemann integrable on [a, b].

• Let f be integrable on [a, b] and ∀x ∈ [a, b], set F (x) =∫ x

af(t)dt. Then F is uniformly continuous on [a, b].

FUNCTION EXTENSIONS:

• f : (a, b)→ R is uniformly continuous on (a, b) ⇐⇒ f can be extended to a continuous function f̄ on [a, b].

WEIERSTRASS APPROXIMATION THEOREM:

• Let f be continuous on I = [a, b]. Then ∀ε > 0, ∃ polynomial p s.t. ∀x ∈ I, |f(x)− p(x)| < ε.

Copyright 2011 Josh Engwer

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References[1] R. G. Bartle, D. R. Sherbert, Introduction to Real Analysis. John Wiley & Sons, New York, NY, 2000.

[2] S. R. Lay, Analysis with an Introduction to Proof. Prentice Hall, Upper Saddle River, NJ, 2005.

[3] A. Mattuck, Introduction to Analysis. Prentice Hall, Upper Saddle River, NJ, 1999.

Copyright 2011 Josh Engwer

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