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Calculus 1 Problems & Solutions – Chapter 1 – Section 1.2.1

1.2.1
Continuity

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Review

1. Basis 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 + ... + a_nxⁿ,

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 discontinuous?