Maret School BC Calculus / Relationship between differentiability and continuity
Continuity and Differentiability. Up to this point, we have used the derivative in some powerful ways. For instance, we saw how critical points (places where the. Differentiability and continuity are the two fundamental concepts of differential you should consider the following relationship between these concepts. CONTINUITY AND DIFFERENTIABILITY Geometrical meaning of continuity. (i) Function f will be continuous at x = c if there is no break.
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. 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. The lake is shaped like a circle of radius 1 kilometer and there is a bridge running across the lake.
- Relationship between differentiability and continuity
You are at the point -1,0 while the phone is at the point. 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. 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.
Continuity and Differentiability
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.
Differentiability and continuity
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. That's one way to think about it.
You could say, well, maybe this is the tangent line right over there, but then why can't I make something like this the tangent line?Continuity and Differentiability
That only intersects at the point C comma zero. And then you could keep doing things like that.