Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 1: Limits And Continuity – Section 1.2.1: Continuity

 

1.2.1
Continuity

 

 

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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:
continuous at
a;
discontinuous at
b, c, and d.

 

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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 – One-Sided Continuity

 

 

 

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

 

A function f is continuous on the interval:

 

i.    (a, b) if it's continuous at every point of (a, b);
ii.    [a, b] if it's continuous on (a, b), right-continuous at a, and left-continuous at b;
iii.   [a, b) if it's continuous on (a, b) and right-continuous at a;
iv.  (a, b] if it's continuous on (a, b) and left-continuous at 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 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

 

i.   Every polynomial is continuous everywhere on R.
ii.  Every rational function is continuous on
R except where its denominator is 0.

 

 

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

 

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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.

 

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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.

 

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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.

 

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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.

 

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5.  Prove that if an even function is right-continuous at x = 0, then it's continuous there.

 

Solution

 

 

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