Sunday, January 24, 2010

Differentiability Implies Continuity

Good day to all.

Today we studied Differentiability Implies Continuity a property that will help in simplifying the process of deciding whether a function is differentiable or continuous.


(*answer at the end of the post along with citation)


Opening up class today we reviewed Continuity at a point, on an interval, and even in a function. Continuity at a point has three requirements; do you remember them?

1. There must exist a value for f(c)

2. There must be a limit for f(x) at x = c

3. The limit must equal the value at x = c

Continuity at an interval is not far off from that: f(x) is continuous on an interval only if it is continuous at each x value in the interval. And for a function to be continuous it must be continuous for every x value in it’s domain.


We then continued reviewing by covering differentiability. We started with the definition of derivative at a point:

We also reviewed the graphical and physical meaning of derivative, which are the slope of the tangent line and the instantaneous rate of change of the function respectively.

Then, like with continuity we reviewed differentiability of f(x): at a point x = c if there is a derivative at x = c; on an interval if it is differentiable for

every x-value in the interval; and in a function if it is differentiable of each x-value in it’s domain.


After we had reviewed the basics needed for the day we dove into the important information of the day. Definition (from the dictionary on my computer) : syllogism |silə,jizəm| Noun deductive reasoning as distinct from induction. Do you remember these from geometry?

Ex. of the positive: If pajamas are flannel then they are comfortable -- P => Q

Ex. of the contrapositive: If pajamas are not comfortable then they are not flannel -- ˜Q => ˜P

Ex. of the inverse: If pajamas are not flannel then they are not comfortable -- ˜P => ˜Q

Ex. of the converse: If pajamas are comfor

table then they are flannel -- Q => P

Think about which ones of these are true, then look at the mathematical examples and ask yourself the same question.

If function f is differentiable at x = c then f is continuous at x = c

Ex: just think of any continuous line, and

it’s derivative

If the function f is not continuous at x = c then f is not differentiable at x = c

Ex: any sort of discontinuity, leap, and infinite, removable… will fill this example

If the function f is not differentiable at x = c then f is not continuous at c

Counter Ex: a cusp is continuous, but not differentiable. The reason it is not differentiable is because there are infinite different derivatives at a cusp.

If the function f is continuous at x = c then it is differentiable at x = c

Counter Ex: again a cusp fits this counter example because it is continuous, but has infinite tangent lines, and thus is not differentiable.

Take a look at these graphs, Blue represents that a function that is not continuous is not differentiable. the reason that this is not continuous is because of the step, the limits from the two different sides are different, thus there is no limit, and it cannot be continuous. The red represents a cusp. The cusp is continuous, there is a limit, and a value, and they are equal, but it is not differentiable because at the cusp there are an infinite number of lines tangent to the tip of the cusp.

On a side note, the easiest way to graph a cusp is to use the absolute values, for example: y = |x| + 1 would be a V-shaped graph with a y intercept of 1. To make it negative you can just add a negative sign in front of the absolute value sign because that will flip the whole thing into the negative.

And what we draw from that experience is that:

Differentiability implies Continuity

If a function f is differentiable at x = c, then f is continuous at x = c

If a function f is not continuous at x = c, then f is not differentiable at x = c

If the equation is :
Is the function continuous at the point x = 2?
Not it is not because when x = 2 the numerator and denominator are zero, and thus there is a hole at x = 2 because 0/0 can be any number.
Is the function differentiable at the point x = 2?
No because the function is not continuous at the point.

*The picture was a statue of Socrates

The next scribe will be (drum roll please): Mc Casper


  1. Hey Flying Slug,

    I really like your post because of the way that you have formatted it and how easy it is to read. I only have one suggestion and that is that maybe for your examples of syllogisms you could use the online graphing tool to show the cusp or leap discontinuity. I think I will do that too for my next scribe post, anyway great job!


  2. I thought it was an awesome post, the use of all of the different colors made it not only easy to read but for some strange reason more fun and interesting, so thanks for that. My critique would be that if you are going to do a quiz question asking who an individual in a picture is you should make sure the name of that person is not in the web address you are siting it with. Maybe instead site it at the bottom with the answer. Other wise easiest post to read thus far.

  3. You have a major mathematical error at the end of your post. One of the properties is misstated.

    It would have been more instructive to use a different example of a syllogism than the one given in class.

    Also, as has been suggested in previous scribe posts, additional worked-out examples of the concept or procedure are what are most helpful to your readers.

    Remember we are writing our textbook day-by-day with these scribe posts. So ask yourself, is this good enough for our textbook?

  4. Hey Flying Slug!
    There are a couple of things I would like to point out:
    - As Beston said, it would be better and more comprehensive if you inserted equations and graphs in your post. Some people are visual learners, and this method would be easier. I recognize that we're all in the process of getting used to this blog, and we'll forget to add something now and then...

    -Secondly, suppose there was a person who wasn't in class, and forgot why there is no derivative at a cusp. For this reason, it would be helpful for that person if the explanation was given in the post.

    -Last but not least, a hint to the "major" error Bru mentioned: "THE LIMIT is the one number you can keep f(x) arbitrarily close to, just by keeping x close to c, but NOT EQUAL to c. -i hope this helps :)

    Flying Slug, I enjoyed reading your post, and I thought the picture of Socrates, and the syllogism about him as well was cool.

  5. oh! I forgot: I have a question about the exploration 4-6B. Question number 7 says : a function is non-differentiable at a point x=c if:
    1. f(C) doesn't exist,
    2. there is a cusp at x=c,
    3. ???
    -> Can anyone help me with the 3rd answer?

    Thank you

  6. Marley, that is my mistake. If you look on the slide from that day's class, I have crossed out number 3.

  7. I like this post. I liked how you explained continuity and differentiability. Those three requirements for continuity were good to have in the post because there are easily forgotten. I get what you mean by differentiability implies continuity. Because in most cases when a function is differentiable it is continuous. Although this post was good, i thought it was a bit crowded and the colors went so far as to distract me some. Good overall job though.

  8. this is a really great help in my homework. thanks a lot. :)