Continuity and Limits

Many theorems in calculus require that functions be continuous on intervals of real
numbers. To successfully carry out
differentiation
and
integration
over an interval, it is important to make sure the function
is continuous.

Definition

A function f is continuous at a point (c, f(c)) if all three conditions are satisfied:

1) An output of c exists:

2) The limit exists for c and

3) The limit equals the output of c

This definition basically means that there is no missing point, gap, or split for
f(x) at c. In other words, you can move your pencil along the image of the function
and you would not have to lift up the pencil. These functions are called smooth
functions.


Continuous function on [a,b]

To see if the three conditions of the definition are satisfied is a simple process.

1) Plug in the value assigned to c into the function and see if f(c) exists.

2) Use the limit definition to see if the limit exists as x approaches c.

The limit is the same coming from the left and from the right of f(c)

3) If the limit exists, see if it is the same as f(c). If it is all of the
above, it is continuous.

We can see that functions need to be continuous in order to be differentiable. Are
all continuous functions differentiable?

The answer is no. In
taking the derivative
we did an example of a continuous function that was
not differentiable at x = 0.


f(x) = |x| is a continuous function but it is not differentiable at x = 0. Even
though it is continuous and we can draw the graph without lifting our pencil, it
is not differentiable. Conversely, all differentiable functions are continuous.

Discontinuous Graphs

There are three types of discontinuities – infinite discontinuities, jump
discontinuities, and point discontinuities.



Infinite Discontinuity

Infinite discontinuities break the 1st condition: They have an asymptote instead
of a specific f(c) value.


Jump Discontinuity

Jump discontinuities break the 2nd condition: The limit approaching from a specific
c from the left is not the same as the limit approaching c from the right.


Point Discontinuity

Point discontinuities break the 3rd condition: The limit of c is not the same as
(c).

These graphs are discontinuous because they cannot be drawn without lifting up the
pencil.
Discontinuous graphs can be differentiated and integrated, but only over a continuous
interval of the graph.

Intermediate Value Theorem

The Intermediate value theorem states that if we have a continuous
function f(x) on the interval [a,b] with M being any number between f(a)
and f(b), there exists a number c such that:

1) a < c=""><>

2) f(c) = M.

The Intermediate
Value Theorem is a geometrical application illustrating that continuous
functions will take on all values between f(a) and f(b). We can see if we draw a horizontal line from M, it will hit the graph at least once. If the function is not continuous on the interval, this theorem would not hold.

It is important to note that this theorem does not tell us the value of M, but only that it exists. For example, we can use this theorem to see if a function will have any x intercepts.

(1) Use the Intermediate Value theorem to determine if f(x) =
2x3 – 5x<> – 10x + 5
has a root somewhere in the
interval [-1,2].

In other words, we are asking if f(x) = 0 in the interval [-1,2]. Using the theorem, we can say that we want to show that there is a number c where -1 < c="">< 2="" such="" that="" m="0" in="" between="" f(-1)="" and="">

We see that p(-1) = 8 and p(2) = -19. Therefore, 8 > 0 > -19, and at least one root exists for f(x).

Similarly to the concept of a limit, it is
important to develop an intuitive understanding of continuity and what
it means in terms of limits. By taking infinitesimally close values of x
(the domain), we can make each f(x) as close as we want. We should also
have a geometric understanding of continuous functions (Intermediate Value Theorem).

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