# Continuity, Intermediate Value Theorem (2.3)

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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