discontinuities - math 464/506, real...
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DiscontinuitiesMATH 464/506, Real Analysis
J. Robert Buchanan
Department of Mathematics
Fall 2007
J. Robert Buchanan Discontinuities
One-sided Continuity
Definition (Continuity from the left)
Suppose f : D(f ) → R and x0 ∈ D(f ). Then f is continuousfrom the left at x0 if ∀ǫ > 0, ∃δ > 0 ∋ ∀x ∈ D(f ),x0 − δ < x < x0 ⇒ |f (x) − f (x0)| < ǫ.
Definition (Continuity from the right)
Suppose f : D(f ) → R and x0 ∈ D(f ). Then f is continuousfrom the right at x0 if ∀ǫ > 0, ∃δ > 0 ∋ ∀x ∈ D(f ),x0 < x < x0 + δ ⇒ |f (x) − f (x0)| < ǫ.
J. Robert Buchanan Discontinuities
Sequential Criterion
Theorem (Sequential criterion for one-sided continuity)
(a) A function f : D(f ) → R is continuous from the left at apoint x0 ∈ D(f ) iff for all sequences {xn} inD(f ) ∩ (−∞, x0) ∋ xn → x0, f (xn) → f (x0).
(b) A function f : D(f ) → R is continuous from the right at apoint x0 ∈ D(f ) iff for all sequences {xn} inD(f ) ∩ (x0,+∞) ∋ xn → x0, f (xn) → f (x0).
Theorem
Suppose f : D(f ) → R and x0 ∈ D(f ), then f is continuous at x0
iff f is both continuous from the left at x0 and continuous fromthe right at x0.
J. Robert Buchanan Discontinuities
Sequential Criterion
Theorem (Sequential criterion for one-sided continuity)
(a) A function f : D(f ) → R is continuous from the left at apoint x0 ∈ D(f ) iff for all sequences {xn} inD(f ) ∩ (−∞, x0) ∋ xn → x0, f (xn) → f (x0).
(b) A function f : D(f ) → R is continuous from the right at apoint x0 ∈ D(f ) iff for all sequences {xn} inD(f ) ∩ (x0,+∞) ∋ xn → x0, f (xn) → f (x0).
Theorem
Suppose f : D(f ) → R and x0 ∈ D(f ), then f is continuous at x0
iff f is both continuous from the left at x0 and continuous fromthe right at x0.
J. Robert Buchanan Discontinuities
Some Types of Discontinuities
Definition
If limx→x0
f (x) exists but either f (x0) does not exist or
limx→x0
f (x) 6= f (x0), then we say that f has a removable
discontinuity at x0.
Definition
If limx→x−
0
f (x) and limx→x+
0
f (x) both exist but limx→x−
0
f (x) 6= limx→x+
0
f (x),
then we say that f has a jump discontinuity at x0.
J. Robert Buchanan Discontinuities
Some Types of Discontinuities
Definition
If limx→x0
f (x) exists but either f (x0) does not exist or
limx→x0
f (x) 6= f (x0), then we say that f has a removable
discontinuity at x0.
Definition
If limx→x−
0
f (x) and limx→x+
0
f (x) both exist but limx→x−
0
f (x) 6= limx→x+
0
f (x),
then we say that f has a jump discontinuity at x0.
J. Robert Buchanan Discontinuities
Discontinuities (cont.)
Definition
A function f is said to have a simple discontinuity (or adiscontinuity of the first kind ) at x0 if f has either aremovable discontinuity or a jump discontinuity at x0. Any otherdiscontinuity of f at x0 is called an essential discontinuity (ora discontinuity of the second kind ).
Definition
(a) A function f is said to have an infinite discontinuity at x0
if either limx→x−
0
f (x) or limx→x+
0
f (x) is infinite.
(b) Any other discontinuity of the second kind is called anoscillating discontinuity .
J. Robert Buchanan Discontinuities
Discontinuities (cont.)
Definition
A function f is said to have a simple discontinuity (or adiscontinuity of the first kind ) at x0 if f has either aremovable discontinuity or a jump discontinuity at x0. Any otherdiscontinuity of f at x0 is called an essential discontinuity (ora discontinuity of the second kind ).
Definition
(a) A function f is said to have an infinite discontinuity at x0
if either limx→x−
0
f (x) or limx→x+
0
f (x) is infinite.
(b) Any other discontinuity of the second kind is called anoscillating discontinuity .
J. Robert Buchanan Discontinuities