GOD AND CALCULUS
relationships between the concepts are fully determined by axioms . definitions of continuity, derivative and differentiability were given to the. of functions, continuity, differentiability, sequences and series of functions, and. Riemann integration. . In this chapter, we define sets, functions, and relations and discuss some of “God created the integers and the rest is the work of man. The idea of continuity rules out breaks, jumps, or holes by demanding that the behavior of One of the examples showing that his faith in God was present was Moses when he led the . A function f is said to be differentiable at x if and only if.
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.
Differentiability and continuity (video) | Khan Academy
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.
- Differentiability and continuity
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.
Relationship between differentiability and continuity
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.
general topology - differentiability and continuity - Mathematics Stack Exchange
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. Calculus is one of the fundamental mathematical tools used for this purpose. Calculus was invented to answer questions that could not be solved by using algebra or geometry. One branch of calculus, called Differential calculus, begins with a question about the speed of moving objects. For example, how fast does a stone fall two seconds after it has been dropped from a cliff? The other branch of calculus, Integral calculus, was invented to answer a very different kind of question: Although these branches began by solving different problems, their methods are the same, since they deal with the rate of change.
Some anticipations of calculus can be seen in Euclid and other classical writers, but most of the ideas appeared first in the seventeenth century. Sir Isaac Newton — and Gottfried W. Leibniz — independently discovered the fundamental theorem of calculus. After its start in the seventeenth century, calculus went for over a century without a proper axiomatic foundation. Newton wrote that calculus could be rigorously founded on the idea of limits, but he never presented his ideas in detail.
A limit, roughly speaking, is the value approached by a function near a given point. During the eighteenth century many mathematicians based their work on limits, but their definition of limit was not clear. InJoseph Louis Lagrange at the Berlin Academy proposed a prize for a successful axiomatic foundation for calculus. He and others were interested in being as certain of the internal consistency of calculus as they were about algebra and geometry.
No one was able to successfully respond to the challenge. It remained for Augustin Louis Cauchy to show, sometime aroundthat the limits can be defined rigorously by means of inequalities Hughes-Hallett, Purpose of the study Calculus is one of the subjects being taught for higher mathematics in high schools and colleges.
The purpose of this paper is to show how to use calculus in our relationship with God. According to Avery J.
Thompson, "Any credence given to the study of mathematics must recognize that God is the original mathematician. And though, through the ages, humankind has experimented to be able to draw conclusion in the areas of mathematics, God's laws are error-free and constant. His everlasting watch-care in the 'natural' cyclic phenomena of this earth daily proves His mathematical supremacy. Galileo is remembered for having acknowledged that 'mathematics is the language that God used to create the universe'".