Calculus Of One Real Variable By Pheng Kim Ving

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.

Fig.
1.1
f is:

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2. Continuity 
Definition 2.1
Continuity At A Point
A function f is said to be continuous at a point a if: 
Definition 2.2 OneSided Continuity

In Fig. 1.1, f is leftcontinuous at c but not rightcontinuous there.
Remark 2.1
A function f is
continuous at a point a
iff it's right and leftcontinuous there. This property follows from the
property that
f has the limit at a iff it has
equal righthand and lefthand limits there.
Definition 2.3 Continuity On An Interval
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


Fig. 3.1
Composition of functions f and g.

Proof
The proofs for parts iiii 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 + 3x^{2}. 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 nonnegative integer
power of x, ie:
P(x) = a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n},
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
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.
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.
Problems & Solutions 
1. Let:
Where is f continuous and where is it discontinuous?
One root of x^{3} 7x + 6 = 0 is x = 1. Let's do long division of x^{3} 7x + 6 by x 1:
So:
x^{3} 7x + 6 = (x 1)(x^{2} + 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) = (x^{2} 1)/x^{2} 1. Where is f continuous and where is it discontinuous? At points of
discontinuity if any, explain why
it's discontinuous.
Solution
If x^{2} 1 > 0 or x > 1, then x^{2} 1 = x^{2} 1, so f (x) = 1. If x^{2} 1 < 0 or x < 1, then x^{2} 1 = (x^{2} 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?
a.
b. If
a is a noninteger 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 noninteger and
discontinuous at
every integer.
5. Prove that if an even function is rightcontinuous at x = 0, then it's continuous there.
Solution
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