Section 4 – 2A: Continuity and Differentiability ...

Section 4 – 2A: Continuity and Differentiability

Continuity

Many theorems in this class use the phrase “if the functions is continuous” which implies the function is
continuous for all x vales. Other theorems use the phrase “if the functions is continuos in an interval (a,b)”
which implies the function is continuous for all x vales between x = a and x = b. In either case we ned to
discuss what this means.

Continuity at a point

lim lim
x → c− f (x) = x → c + f (x) = f (c)

This definition requires that the point (c, f (c)) has a y value at x = c and that the y values of the points on

the right and left sides of the point approach f (c) as x approaches c. In other words, the y values of the
function on the right and left sides of the must both approach the y value of the point which is f (c)

Continuity on an open Interval

A function is said to be continuous on an open interval (a,b) when the function is defined at every point on
the closed [a,b] interval and undergoes no interruptions, jumps, or breaks on the interval [a,b].
If the domain is limited to the closed interval [a,b] then the function cannot be continuous at the endpoints.

The definition of continuity at a point requires a limit on both side of a point . The left endpoint does not
have a left sided limit and the right endpoint does not have a right sided limit because of the domain

restriction so the function is not continuous on the closed interval [a,b]. SInce every point on the
open interval (a,b) has a left and right sided limit we can have a function is continuous on the open
interval (a,b) if it is continuous at every point on the interval.

The function is defined on [a,b] The function is NOT defined on [a,b]

and has no no interruptions, jumps, because there are no y values for x
at x = a and x =b
or breaks on [a,b]
y
y

x=a x=b x=a x=b

The function is continuous The function is not continuous

on an open interval (a,b) on an open interval (a,b)

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A general concept of Continuity

We only require a “general” description of what we mean by continuity for this chapter. A function is
continuous on an open interval (a,b) if there are no breaks in the graph between x = a and x = b. If the
graph has any holes, vertical asymptotes, gaps in the range or any domain restrictions in the
interval (a,b) then the function is discontinuous on the interval.

A function is discontinuous on the open interval (a,b) if:

There is a hole in the There is a vertical asymptote
graph at x = c in the graph at x = c
y
y

x=a x=c x x=a x
x=b x=c x=b

There is gap in the y values. There There is domain restriction. No values of x
is a gap in the graph at x = c are defined for x values to the left of x = c

y y

x=a x=c x x=a x=c x=b
x=b

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Differentiability

Many theorems in this class use the phrase “if the functions is differentiable”. This does not mean you can
take the derivative of the function. It means that when you do take the derivative of the function the
derivative is defined for every value of x . There are no values of x where the derivative is undefined.
The most common way for a derivative to be undefined is for the derivative to have an expression in the
denominator that can equal 0 for some x value. A second way for the derivative to be undefined is for a
range of x values are are not allowed based on a domain restriction from a radical expression or logarithmic
function. Another way for the derivative to be undefined is for the derivative to be expressed in set
builder format and the graph of that has a hole or gap in it.

Example 1 Example 2
y = x 2/3 is defined for all values of x.
y = x + 2
x

y′ = 2 3 y′ = −2
3 x1 x2

y′ is not defined for x = 0 y′ is not defined for x = 0

y′ is not differentiable for x = 0 y′ is not differentiable for x = 0

y′ is not differentiable for any y′ is not differentiable for any
interval that containes x = 0 interval that containes x = 0

What is the difference between Example 1 and Example 2

In both cases the derivative is not defined at x = 0

In Example 1 the function is defined at x = 0. f (0) = 0 so there is a point on the graph at (0,0). The
derivative is not defined at x = 0. The slope of the tangent line at the point (0,0) cannot be found. There
is a point at (0,0) but we cannot state the slope at that point. We will discuss why this is true in a later

section.

In Example 2 the function is NOT defined at x = 0. This is a domain restriction on the original function. The

original function is not defined at f (0) so there is NOT a point on the graph at x = 0. In fact there is a vertical

asymptote at x = 0. The derivative is also not defined at x = 0. Since there is no point at x = 0 the
question of slope makes no sense.

Note: any domain restriction on the original function always carries over to any derivative of that function.
If a function is not defined at x = c then no derivative of that function will be defined at x = c

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What is the connection between Continuity and Differentiability

If a function is differentiable in an interval ( a,b) then it is continuous on the an interval ( a,b)

If a function is differentiable in an interval ( a,b) then there are no x values of x where the derivative is
undefined. If a function is differentiable in an interval ( a,b) then the graph of the original function is either a
line or a smooth curve. The graph of the original function has no gaps, holes vertical asymptotes, domain
restrictions or sharp points on it. This implies the function is continuous on the interval ( a,b)

The converse statement is NOT true. If a function is continuous on the an interval ( a,b) there is no
guarantee that the functions derivative is defined for all values of x on the interval

Differentiability implies Continuity
but Continuity DOES NOT imply Differentiability

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