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8/7/2019 Continuity Slides
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Definition: Let D (= ∅) ⊂ R and let f : D → R.
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Definition: Let D (= ∅) ⊂ R and let f : D → R.
We say that f is continuous at x 0 ∈ D if for each ε > 0, there
exists δ > 0 such that |f (x ) − f (x 0)| < ε for all x ∈ D satisfying
|x − x 0| < δ.
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Definition: Let D (= ∅) ⊂ R and let f : D → R.
We say that f is continuous at x 0 ∈ D if for each ε > 0, there
exists δ > 0 such that |f (x ) − f (x 0)| < ε for all x ∈ D satisfying
|x − x 0| < δ.
We say that f : D → R is continuous if f is continuous at each
x 0 ∈ D .
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Definition: Let D (= ∅) ⊂ R and let f : D → R.
We say that f is continuous at x 0 ∈ D if for each ε > 0, there
exists δ > 0 such that |f (x ) − f (x 0)| < ε for all x ∈ D satisfying
|x − x 0| < δ.
We say that f : D → R is continuous if f is continuous at each
x 0 ∈ D .
Definition: Let D ⊂ R and let x 0 ∈ R such that for some h > 0,
(x 0 − h , x 0 + h ) \ {x 0} ⊂ D .
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Definition: Let D (= ∅) ⊂ R and let f : D → R.
We say that f is continuous at x 0 ∈ D if for each ε > 0, there
exists δ > 0 such that |f (x ) − f (x 0)| < ε for all x ∈ D satisfying
|x − x 0| < δ.
We say that f : D → R is continuous if f is continuous at each
x 0 ∈ D .
Definition: Let D ⊂ R and let x 0 ∈ R such that for some h > 0,
(x 0 − h , x 0 + h ) \ {x 0} ⊂ D .
If f : D → R, then ∈ R is said to be the limit of f at x 0 if for
each ε > 0, there exists δ > 0 such that |f (x ) − | < ε for all
x ∈ D satisfying 0 < |x − x 0| < δ.
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Definition: Let D (= ∅) ⊂ R and let f : D → R.
We say that f is continuous at x 0 ∈ D if for each ε > 0, there
exists δ > 0 such that |f (x ) − f (x 0)| < ε for all x ∈ D satisfying
|x − x 0| < δ.
We say that f : D → R is continuous if f is continuous at each
x 0 ∈ D .
Definition: Let D ⊂ R and let x 0 ∈ R such that for some h > 0,
(x 0 − h , x 0 + h ) \ {x 0} ⊂ D .
If f : D → R, then ∈ R is said to be the limit of f at x 0 if for
each ε > 0, there exists δ > 0 such that |f (x ) − | < ε for all
x ∈ D satisfying 0 < |x − x 0| < δ.
We write: limx →x 0
f (x ) = .
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Similarly we define: limx →x 0+
f (x ) = and limx →x 0−
f (x ) = .
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Similarly we define: limx →x 0+
f (x ) = and limx →x 0−
f (x ) = .
Result: Let D ⊂ R and let x 0 ∈ D such that for some h > 0,
(x 0 − h , x 0 + h ) ⊂ D . Then f : D → R is continuous iff
limx →x 0
f (x ) = f (x 0).
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Similarly we define: limx →x 0+
f (x ) = and limx →x 0−
f (x ) = .
Result: Let D ⊂ R and let x 0 ∈ D such that for some h > 0,
(x 0 − h , x 0 + h ) ⊂ D . Then f : D → R is continuous iff
limx →x 0
f (x ) = f (x 0).
Similarly the other two cases.
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Similarly we define: limx →x 0+
f (x ) = and limx →x 0−
f (x ) = .
Result: Let D ⊂ R and let x 0 ∈ D such that for some h > 0,
(x 0 − h , x 0 + h ) ⊂ D . Then f : D → R is continuous iff
limx →x 0
f (x ) = f (x 0).
Similarly the other two cases.
Sequential criterion of continuity: f : D → R is continuous at
x 0 ∈ D iff for every sequence (x n ) in D such that x n → x 0, we
have f (x n ) → f (x 0).
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Similarly we define: limx →x 0+
f (x ) = and limx →x 0−
f (x ) = .
Result: Let D ⊂ R and let x 0 ∈ D such that for some h > 0,
(x 0 − h , x 0 + h ) ⊂ D . Then f : D → R is continuous iff
limx →x 0
f (x ) = f (x 0).
Similarly the other two cases.
Sequential criterion of continuity: f : D → R is continuous at
x 0 ∈ D iff for every sequence (x n ) in D such that x n → x 0, we
have f (x n ) → f (x 0).
Similar criterion for limit.
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Examples:
1. f (x ) =
3x + 2 if x < 1,4x 2 if x ≥ 1.
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Examples:
1. f (x ) =
3x + 2 if x < 1,4x 2 if x ≥ 1.
2. f (x ) =
x sin 1
x if x = 0,
0 if x = 0.
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Examples:
1. f (x ) =
3x + 2 if x < 1,4x 2 if x ≥ 1.
2. f (x ) =
x sin 1
x if x = 0,
0 if x = 0.
3. f (x ) =
sin1
x if x = 0,0 if x = 0.
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Examples:
1. f (x ) =
3x + 2 if x < 1,4x 2 if x ≥ 1.
2. f (x ) =
x sin 1
x if x = 0,
0 if x = 0.
3. f (x ) =
sin1
x if x = 0,0 if x = 0.
4. f (x ) =
1 if x ∈ Q,0 if x ∈ R \Q.
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Examples:
1. f (x ) =
3x + 2 if x < 1,4x 2 if x ≥ 1.
2. f (x ) =
x sin 1
x if x = 0,
0 if x = 0.
3. f (x ) =
sin1
x if x = 0,0 if x = 0.
4. f (x ) =
1 if x ∈ Q,0 if x ∈ R \Q.
5. f (x ) =
x if x ∈ Q,−x if x ∈ R \Q.
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Result: Let f ,g : D → R be continuous at x 0 ∈ D . Then
(i) f + g , fg and |f | are continuous at x 0
,
(ii) f /g is continuous at x 0 if g (x 0) = 0.
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Result: Let f ,g : D → R be continuous at x 0 ∈ D . Then
(i) f + g , fg and |f | are continuous at x 0
,
(ii) f /g is continuous at x 0 if g (x 0) = 0.
Ex. Similar results for discontinuous functions?
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Result: Let f ,g : D → R be continuous at x 0 ∈ D . Then
(i) f + g , fg and |f | are continuous at x 0
,
(ii) f /g is continuous at x 0 if g (x 0) = 0.
Ex. Similar results for discontinuous functions?
Ex. If f : D → R is continuous at x 0 and if f (x 0) = 0, then show
that there exists δ > 0 such that f (x ) = 0 for allx ∈ (x 0 − δ,x 0 + δ).
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Result: Let f ,g : D → R be continuous at x 0 ∈ D . Then
(i) f + g , fg and |f | are continuous at x 0
,
(ii) f /g is continuous at x 0 if g (x 0) = 0.
Ex. Similar results for discontinuous functions?
Ex. If f : D → R is continuous at x 0 and if f (x 0) = 0, then show
that there exists δ > 0 such that f (x ) = 0 for allx ∈ (x 0 − δ,x 0 + δ).
Result: Composition of two continuous functions is continuous.
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Result: Let f ,g : D → R be continuous at x 0 ∈ D . Then
(i) f + g , fg and |f | are continuous at x 0
,
(ii) f /g is continuous at x 0 if g (x 0) = 0.
Ex. Similar results for discontinuous functions?
Ex. If f : D → R is continuous at x 0 and if f (x 0) = 0, then show
that there exists δ > 0 such that f (x ) = 0 for allx ∈ (x 0 − δ,x 0 + δ).
Result: Composition of two continuous functions is continuous.
Further examples of continuous functions:
Polynomial function, Rational function, sine function, cosine
function, etc.
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Result: If f : [a ,b ] → R is continuous and if f (a ) · f (b ) < 0, then
there exists c ∈ (a ,b ) such that f (c ) = 0.
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Result: If f : [a ,b ] → R is continuous and if f (a ) · f (b ) < 0, then
there exists c ∈ (a ,b ) such that f (c ) = 0.
Intermediate value theorem: Let I be an interval of R and let
f : I → R be continuous. If a ,b ∈ I with a < b and if
f (a ) < k < f (b ), then there exists c ∈ (a ,b ) such that f (c ) = k .
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Result: If f : [a ,b ] → R is continuous and if f (a ) · f (b ) < 0, then
there exists c ∈ (a ,b ) such that f (c ) = 0.
Intermediate value theorem: Let I be an interval of R and let
f : I → R be continuous. If a ,b ∈ I with a < b and if
f (a ) < k < f (b ), then there exists c ∈ (a ,b ) such that f (c ) = k .
Examples:
1. The equation x 2 = x sin x + cos x has at least two real roots.
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Result: If f : [a ,b ] → R is continuous and if f (a ) · f (b ) < 0, then
there exists c ∈ (a ,b ) such that f (c ) = 0.
Intermediate value theorem: Let I be an interval of R and let
f : I → R be continuous. If a ,b ∈ I with a < b and if
f (a ) < k < f (b ), then there exists c ∈ (a ,b ) such that f (c ) = k .
Examples:
1. The equation x 2 = x sin x + cos x has at least two real roots.
2. If f : [0, 1] → [0, 1] is continuous, then there exists x 0 ∈ [0, 1]such that f (x 0) = x 0.
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Result: If f : [a ,b ] → R is continuous and if f (a ) · f (b ) < 0, then
there exists c ∈ (a ,b ) such that f (c ) = 0.
Intermediate value theorem: Let I be an interval of R and let
f : I → R be continuous. If a ,b ∈ I with a < b and if
f (a ) < k < f (b ), then there exists c ∈ (a ,b ) such that f (c ) = k .
Examples:
1. The equation x 2 = x sin x + cos x has at least two real roots.
2. If f : [0, 1] → [0, 1] is continuous, then there exists x 0 ∈ [0, 1]such that f (x 0) = x 0.
3. If f : [0, 2] → R is continuous such that f (0) = f (2), then
there exist x 1, x 2 ∈ [0, 2] such that x 1 − x 2 = 1 andf (x 1) = f (x 2).
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Result: If f : [a ,b ] → R is continuous, then f : [a ,b ] → R is
bounded.
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Result: If f : [a ,b ] → R is continuous, then f : [a ,b ] → R is
bounded.
Result: If f : [a ,b ] → R is continuous, then there exist
x 0, y 0 ∈ [a ,b ] such that f (x 0) ≤ f (x ) ≤ f (y 0) for all x ∈ [a ,b ].
