Continuity, Intermediate Value Theorem (2.3)

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Continuity, Intermediate Value Theorem (2.3)

Transcript Of Continuity, Intermediate Value Theorem (2.3)

Continuity Theorems about Limits

Continuity, Intermediate Value Theorem (2.3)
Xiannan Li
Kansas State University
August 31st, 2017

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

Intuitive Definition: A function f (x) is continuous at a if you can draw the graph of y = f (x) without lifting your pen when x = a.
Definition (continuity) A function f (x) is continuous at a number a if
lim f (x) = f (a).
x→a
This means that a function f (x) is continuous at a provided that:
1 f (a) is defined 2 lim f (x) exists
x→a
3 lim f (x) = f (a)
x→a

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

Definition (continuity) A function f (x) is continuous at a number a if
lim f (x) = f (a).
x→a

Each of the plots above are discontinuous at x = 2 but continuous at x = 1.

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

Classification: We classify discontinuities of a function f (x) as one of removable, jump, infinite or other.

removable disc.

jump disc.

infinite disc.

Each of the plots above are discontinuous at x = 2. 1 For removable discontinuities, the limit exists, but is not equal to the function value. 2 For jump discontinuities, the left and right limits exist but are not equal (so the limit does not exist). 3 For infinite discontinuities, left or right limits are ±∞..

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

Classification: We classify discontinuities of a function f (x) as one of removable, jump, infinite or other.

lim f (x), lim f (x), and lim f (x) do not exist

x→0−

x→0+

x→0

This familiar plot is discontinuous at x = 0. This oscillating discontinuity is classified as ”other”.

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

We say that a function f (x) is continuous on an interval if it is continuous at every point in the interval.

Example: Where is f (x) = x−1 2 continuous/discontinuous?

Continuous:

Discontinuous:

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

Definition 1 A function f (x) is continuous from the left at a number a if
lim f (x) = f (a).
x→a−
2 A function f (x) is continuous from the right at a number a if lim f (x) = f (a).
x→a+
Note, f (x) is continuous at a if and only if f (x) is continuous from both the right and left at a.

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

The function above is discontinous at x = 2, continuous from the right at x = 2, and discontinuous from the left at x = 2.
The function is continuous on (0, 2) and (2, 4).

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

From earlier in the lecture, we know that wherever they are

defined, polynomials, rational functions, and root functions are

all continuous. This is what we really were using when we said

for example that

lim (x2 − 3x + 1) = −1.
x→2

Theorem

The following types of functions are continuous at every number

in their domains:

polynomials (ex., x3 + x + 1)

rational functions (e√x., x3x+2+x+1 1 ) root functions (ex., 3 x)

trigonometric functions (ex., cos(x))

inverse trigonometric functions (ex., arcsin(x)) exponential functions (ex., ex)

logarithmic functions (ex., ln(x))

Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)

Continuity Theorems about Limits

Definitions Properties

Theorem If f (x) and g(x) are continuous at a and c is a constant, then the following functions are also continuous at a:
f (x) + g(x) f (x) − g(x) c · f (x) f (x) · g(x) fg((xx)) if g(a) = 0.

Applying these rules multiple times, we see that

√ 3x

+

x47

cos(x)

h(x) =

ex

is continuous wherever it is defined (which is everywhere).

Thus, lim h(x) =
x→3
Math 220 Lecture 4

Continuity, IVT (2.3 & 2.4)
FunctionsDiscontinuitiesContinuityPolynomialsInterval