Continuity and Differentiability
Definition of continuity at a point. A function f(x) Definition of continuity in an interval. Let 'h' be a Relation between continuity and differentiability. (1) if f(x) is . However, continuity and differentiability are included among the most difficult differentiability implies continuity, we prove the second relation using the first. Explanation: 1) If a function is differentiable, then by definition of differentiability the limit defined by,. exists. Therefore (1) is required by definition of.
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.
Differentiability and continuity
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. The lake is shaped like a circle of radius 1 kilometer and there is a bridge running across the lake. You are at the point -1,0 while the phone is at the point.
Let's find out how long it would take to reach the phone if you first swam across the lake at an angle t and then ran along the shore to the phone. Remember that distance is equal to velocity times time so that the length of time it takes to swim is equal to the distance that you swim divided by the velocity at which you swim.
Differentiability and continuity (video) | Khan Academy
To compute the distance you swimwe can use the law of cosines. It says that In other words, so that the time it takes you to swim is. Now, once you reach the opposite shore, you still have to run a distance along the shore. The phone is at a position which corresponds to a central angle of. Since you finish your swim at a position which corresponds to a central angle ofthe distance you must run is The amount of time it takes to do this is which says that the total time it takes to reach the phone along this route is Of course, we have to be careful since this formula assumes we are actually swimming for some of the time.
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.
Relationship between differentiability and continuity
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.
- Differentiability and its relation with continuity
- Continuity and Differentiability
And so as we approach from the left, as X approaches C from the left, we actually have a situation where this expression right over here is going to approach negative infinity. And if we approach from the right, if we approach with Xs larger than C, well, this is our X comma F of X, so we have a positive slope and then as we get closer, it gets more positive, more positive approaches positive infinity. But either way, it's not approaching a finite value. And one side is approaching positive infinity, and the other side is approaching negative infinity.
This, the limit of this expression, is not going to exist. So once again, I'm not doing a rigorous proof here, but try to construct a discontinuous function where you will be able to find this. It is very, very hard. And you might say, well, what about the situations where F is not even defined at C, which for sure you're not gonna be continuous if F is not defined at C.
Well if F is not defined at C, then this part of the expression wouldn't even make sense, so you definitely wouldn't be differentiable. But now let's ask another thing. I've just given you good arguments for when you're not continuous, you're not going to be differentiable, but can we make another claim that if you are continuous, then you definitely will be differentiable?
Well, it turns out that there are for sure many functions, an infinite number of functions, that can be continuous at C, but not differentiable.
So for example, this could be an absolute value function. It doesn't have to be an absolute value function, but this could be Y is equal to the absolute value of X minus C. And why is this one not differentiable at C? Well, think about what's happening. Think about this expression. Remember, this expression all it's doing is calculating the slope between the point X comma F of X and the point C comma F of C. So if X is, say, out here, this is X comma F of X, it's going to be calculated, so if we take the limit as X approaches C from the left, we'll be looking at this slope.
And as we get closer, we'll be looking at this slope which is actually going to be the same. In this case it would be a negative one.
So as X approaches C from the left, this expression would be negative one. But as X approaches C from the right, this expression is going to be one. The slope of the line that connects these points is one. So the limit of this expression, or I would say the value of this expression, is approaching two different values as X approaches C from the left or the right. From the left, it's approaching negative one, or it's constantly negative one and so it's approaching negative one, you could say.
And from the right, it's one, and it's approaching one the entire time. And so we know if you're approaching two different values from on the left side or the right side of the limit, then this limit will not exist. So here, this is not, not differentiable. And even intuitively, we think of the derivative as the slope of the tangent line.
And you could actually draw an infinite number of tangent lines here.