Continuity

Abstract

  • Continuity criterionThat is, no matter how close we make the y-coordinates, we can always find two x-coordinates that make these two y-coordinates close enough.
  • All polynomials, rational functions, trig functions, exponential functions, logarithmic function, and root functions are continuous on their domains
  • is continuous at
    • Look for the limit (may need both sides) and then verify that the limit is equal to
  • If two functions are continuous, then their composition, addition, multiplication and division are continuous also
  • IVT:
    • With continuous on , if is between and , then
    • If you really think about this, it's quite intuitive. If there's a number between the two y-coordinates, you can find an x-coordinate to match it in that interval.
  • EVT:
    • is continuous on attains a max and min value.

The Continuity Criterion

Definition

A function is continuous at if for all , there exists a such that:

This tells you

That is, we can always force to be close to by making close enough to

A function is said to be continuous if it is continuous at every point in the domain

  1. See implication, The Epsilon Delta Definition
  2. We don't have at the start, because could equal
Examples

left = -3; right = 3;
bottom = -3; top = 3;
---
y = \frac{1}{x}

The function is continuous on the domain. We can determine this by looking at the graph.

Theorem

is continuous at (see If and Only If)

Two steps:

  1. Look for
    • You may need to look for and
  2. Verify
Theorem

Suppose and are continuous on an interval. Then:

  • is continuous at every point in the interval
  • is continuous on the interval
  • is continuous on the interval
  • is continuous on the interval where
Examples

  • is continuous for all values of
  • is continuous for all values of
  • is continuous for all

Continuity of a Piecewise Function

We know this is continuous for and , but is it continuous at ?

Since the limits on both sides are equal, is continuous everywhere.

Types of Discontinuities

Intermediate Value Theorem (IVT)

Theorem

Suppose is continuous on a closed interval . If is a number between and (i.e, if or ), then there exists at least one number such that

Or in symbolic form,

See implication, logical or

left = -5; right = 5;
bottom = -20; top = 5;
---
y = 0.25x^3
(-4, -16) | point | label: (a, f(a))
(2, 2) | point | label: (b, f(b))
y = -5 | dashed
(-5.15, -5) | label: y = c

In other words, it is impossible to draw a continuous path from to without crossing the line at least once.

Application: Root Finding

The Extreme Value Theorem

Theorem

If is continuous on a closed interval , then attains a maximum value and a minimum value on that interval.

Functions without Extreme Values

left = -5; right = 5;
bottom = -5; top = 5;
---
y = \frac{1}{x} \{-2 < x < 2\} | #2d70b3
(-2, -0.5) | point | #2d70b3
(2, 0.5) | point | #2d70b3

Here, the function isn't bounded on the given interval

left = -5; right = 5;
bottom = -5; top = 5;
---
y = -x^2 + 2 \{-2 < x < 2\} | #2d70b3
(-2, -2) | point | #2d70b3
(2, -2) | point | #2d70b3
(0, 2) | open | #2d70b3

Here, there is no maximum value. If you name a value a larger one can always be found.

left = -5; right = 5;
bottom = -5; top = 5;
---
y = x \{-2 < x < 2\} | #2d70b3
(-2, -2) | open | #2d70b3
(2, 2) | open | #2d70b3

Similar to the previous case