Differentiability and continuity (video) | Khan Academy
In mathematics, differential calculus is a subfield of calculus concerned with the study of the . The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second A differential equation is a relation between a collection of functions and their derivatives. Example 1 Check the continuity of the function f given by f(x) = 2x + 3 at x = 1. Solution Solution We differentiate the relationship directly with respect to x, i.e. Why is it that all differentiable functions are continuous but not all continuous functions are differentiable? A continuous function is a function whose graph is a single unbroken curve. What does this tell us about the relationship between continuous functions and.
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.
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. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.Relation between Continuity and Differentiability- Pre-Calculus - Ch-5.3c - 12th Std NCERT- Edusaral
This also has applications in graph sketching: In higher dimensionsa critical point of a scalar valued function is a point at which the gradient is zero.
The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a " saddle point ", and if none of these cases hold i. Calculus of variations[ edit ] Main article: Calculus of variations One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface.
If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear.
These paths are called geodesicsand one of the simplest problems in the calculus of variations is finding geodesics.
Differentiability and continuity
Find the smallest area surface filling in a closed curve in space. 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.
Differential calculus - Wikipedia
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. 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.