Differentiable function - Wikipedia
Florida State University the relationship between continuity and differentiability .. “for a function to be continuous is the same as being. Willie Fix. Relationship Between Differentiability and Continuity. All differentiable functions are continuous, but not all continuous functions are differentiable. AP Calculus AB: Relationship between differentiability and continuity. Study concepts List which of the following statements must be true about: 1) The limit 1) If a function is differentiable, then by definition of differentiability the limit defined by,. exists. For a function to be continuous at a point we must have: Therefore.
So let's just remind ourselves a definition of a derivative. And there's multiple ways of writing this. For the sake of this video, I'll write it as the derivative of our function at point C, this is Lagrange notation with this F prime. And at first when you see this formula, and we've seen it before, it looks a little bit strange, but all it is is it's calculating the slope, this is our change in the value of our function, or you could think of it as our change in Y, if Y is equal to F of X, and this is our change in X.
And we're just trying to see, well, what is that slope as X gets closer and closer to C, as our change in X gets closer and closer to zero? And we talk about that in other videos. So I'm now going to make a few claims in this video, and I'm not going to prove them rigorously. There's another video that will go a little bit more into the proof direction. But this is more to get an intuition. So I'm saying if we know it's differentiable, if we can find this limit, if we can find this derivative at X equals C, then our function is also continuous at X equals C.
It doesn't necessarily mean the other way around, and actually we'll look at a case where it's not necessarily the case the other way around that if you're continuous, then you're definitely differentiable. But another way to interpret what I just wrote down is, if you are not continuous, then you definitely will not be differentiable.
So let me give a few examples of a non-continuous function and then think about would we be able to find this limit.
Differentiability and continuity (video) | Khan Academy
So the first is where you have a discontinuity. Our function is defined at C, it's equal to this value, but you can see as X becomes larger than C, it just jumps down and shifts right over here. So what would happen if you were trying to find this limit?
Well, remember, all this is is a slope of a line between when X is some arbitrary value, let's say it's out here, so that would be X, this would be the point X comma F of X, and then this is the point C comma F of C right over here. So this is C comma F of C. So if you find the left side of the limit right over here, you're essentially saying okay, let's find this slope. And then let me get a little bit closer, and let's get X a little bit closer and then let's find this slope. And then let's get X even closer than that and find this slope.
And in all of those cases, it would be zero. The slope is zero. So one way to think about it, the derivative or this limit as we approach from the left, seems to be approaching zero. But what about if we were to take Xs to the right? So instead of our Xs being there, what if we were to take Xs right over here? If we get X to be even closer, let's say right over here, then this would be the slope of this line. If we get even closer, then this expression would be the slope of this line.
And so as we get closer and closer to X being equal to C, we see that our slope is actually approaching negative infinity. And most importantly, it's approaching a very different value from the right. This expression is approaching a very different value from the right as it is from the left.
And so in this case, this limit up here won't exist. So we can clearly say this is not differentiable. So once again, not a proof here. I'm just getting an intuition for if something isn't continuous, it's pretty clear, at least in this case, that it's not going to be differentiable. Let's look at another case. Let's look at a case where we have what's sometimes called a removable discontinuity or a point discontinuity. So once again, let's say we're approaching from the left.
This is X, this is the point X comma F of X. Now what's interesting is where as this expression is the slope of the line connecting X comma F of X and C comma F of C, which is this point, not that point, remember we have this removable discontinuity right over here, and so this would be this expression is calculating the slope of that line.
And then if X gets even closer to C, well, then we're gonna be calculating the slope of that line. If X gets even closer to C, we're gonna be calculating the slope of that line.
Here we suspect that the integer values of t are discontinuities of the function since we could not draw this graph without picking up the pen at these points. However, we cannot force the function to be close to 4 by taking values of t close to 1. Notice that, for a function like this, our usual methods from calculus would not be applicable.
That is, if we wanted to find the maximum value on some interval, we would not be able to find it by looking for critical points. Consider the function and remember that the graph looks like: Here the function is not defined at the points and near these points, the function becomes both arbitrarily large and arbitrarily small.
Since the function is not defined at these points, it cannot be continuous.
Differentiability and continuity
Again, if this function arose in a situation which we wanted to optimize, we would have to be careful when applying our usual methods from calculus. There are some situations which present us with a function which has an "unusual" point in fact, we'll see an example of this later on. Here is an example: Again, if we were to apply the methods we have from calculus to find the maxima or minima of this function, we would have to take this special point into consideration.
Mathematicians have made an extensive study of discontinuities and found that they arise in many forms. In practice, however, these are the principle types you are likely to encounter.
Differentiability We have earlier seen functions which have points at which the function is not differentiable. An easy example is the absolute value function which is not differentiable at the origin. Notice that this function has a minimum value at the origin, yet we could not find this value as the critical point of the function since the derivative is not defined there remember that a critical point is a point where the derivative is defined and zero.
A similar example would be the function. Notice that which shows that the derivative does not exist at.
However, this function has a minimum value at. An example To illustrate how to deal with these kinds of situations, here is an example. Suppose that you are on one side of a lake listening to the radio. There is an announcement that you have won a special prize, but you must call the radio station quickly.
The nearest phone is on the other side of the lake and you would like to reach the phone as quickly as possible. The situation is drawn to the left.