Calculus Of One Real Variable By Pheng Kim Ving
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1.2.1 |
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1. Motivation
For The Definitions Of Continuity |
Returning to a where the graph is continuous, we notice that the
graph of f contains the point (a, f(a)) and a piece of it
touches that point from both sides. ( For comparison,
the graph contains the point (c, f(c)) but a piece of it touches that
point from only one side, and contains the point (d, f(d )) but no piece of it touches that point.) See Section
1.1.1 Remarks
2.1 v. This means that:
exists and equals f (a).
All that's just been examined forms the basis for the definition of continuity.
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Fig.
1.1
f is:
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2. Continuity |
Definition 2.1
Continuity At A Point
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A function f is said to be continuous at a point a if:
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Definition 2.2 One-Sided Continuity
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In Fig. 1.1, f is left-continuous at c but not right-continuous there.
Remark 2.1
A function f is
continuous at a point a
iff it's right- and left-continuous there. This property follows from the
property that
f has the limit at a iff it has
equal right-hand- and left-hand limits there.
Definition 2.3 Continuity On An Interval
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A function f is continuous on the interval: i. (a, b) if it's continuous at every
point of (a, b); |
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3. Continuity Of Some Functions |
Theorem 3.1
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Fig. 3.1
Composition of functions f and g.
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Proof
The proofs for parts i-iii follow readily from the corresponding
properties of limits. See Section
1.1.2 Theorems 3.1 and
3.2.
EOP
Polynomials And Rational Functions
An example of a polynomial of degree 2 is 4
2x + 3x2. A polynomial
is a function P(x) that is the sum of a finite
number of terms each of which is a constant multiple of a non-negative integer
power of x, ie:
P(x) = a0 + a1x + a2x2 + ... + anxn,
A rational function is a function of the form R(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials.
Corollary 3.1
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i. Every polynomial is
continuous everywhere on R. |
The proof of this corollary follows at once from the above theorem and the properties of limits.
Example 3.1
Let:
Where is f continuous and where is it discontinuous?
Solution
f is continuous everywhere on R except at x = 1 and x = 1 where it's
undefined and thus discontinuous.
EOS
Problems & Solutions |
1. Let:
Where is f continuous and where is it discontinuous?
Solution
One root of x3 7x + 6 = 0 is x = 1. Let's do long division of x3 7x + 6 by x 1:
So:
x3 7x + 6 = (x 1)(x2 + x 6) = (x 1)(x + 3)(x 2) = (x + 3)(x 1)(x 2).
Thus f is continuous everywhere except at x = 3, 1, and 2 where it's discontinuous.
2.
Let f be defined by f(x) = (x2 1)/|x2 1|. Where is f continuous and where is it discontinuous? At points of
discontinuity if any, explain why
it's discontinuous.
Solution
If x2 1 > 0 or |x| > 1, then |x2 1| = x2 1, so f (x) = 1. If x2 1 < 0 or |x| < 1, then |x2 1| = (x2 1), so f(x) = 1;
f is not defined at x = 1 or x = 1. Hence, f is continuous everywhere except at x = 1 and x = 1. It's discontinuous
at
these two points because it's not defined at any of them.
3. Let g be defined by:
Where is g continuous and where is it discontinuous? At points of discontinuity if any, explain why it's discontinuous.
Solution
Thus, g is also continuous at x = 1. Hence, g is continuous everywhere on the real line.
The graph of g
is the same as
the line y = x 2.
4. Let
[x] denote the greatest integer less than or equal to x.
a. Sketch a graph of y = [x].
b. Where is [x] continuous and where is it discontinous?
Solution
a.
b. If
a is a non-integer then n < a < n + 1 for some integer n. And:
and thus [x] cannot be
continuous at n. We've seen that [x] is continuous at every non-integer and
discontinuous at
every integer.
5. Prove that if an even function is right-continuous at x = 0, then it's continuous there.
Solution
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