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.
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.
There are three types of discontinuities - infinite discontinuities, jump discontinuities, and point discontinuities.
Infinite discontinuities break the 1st condition: They have an asymptote instead of a specific f(c) value.
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 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 < b
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
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 f(2).
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).