Differentiability and Continuity

[hasan_researc]

Hi,

I don't understand why mathematicians would need to define the mathematical concepts of diffferentiabilty and conitnuity. To be honest, I don't even understand why "f(x) tends to f(a) as x tends to a" describes continuity.

Also, I am wondering why f(x) = mod x is not differentiable at the origin. Surely, f(x) can have two tangents with opposite orientations at the origin, can't it??

Please help!

 

[Gib Z]

We'll Mathematicians like to be very precise to make sure everything they "prove" it actually proven and not just probably true. That's why in the last couple centuries it has focused more on rigor than before, and Clearly stating precise definitions is part of that. We need to know exactly what something is, before we can go and say other things have that property.

In this case, we need to define Differentiability and Continuity because we want to be able to go on and say that "So and so is Differentiable/Continuous, and because they also satisfy something else, we know that they are also Analytic or something". We can't go off vague notions like "something is continuous if we can draw its graph without taking out pen off the page" because there always ends up being inconsistencies, or things not accounted for in those types of definitions.

The first thing for you to understand extremely well is the definition of a Limit. It is Core to the study of Calculus, and you must understand it perfectly before you go on. Then other things can follow on from that relatively easily. The Definitions of continuity and differentiability are simple if you know limits well. And it will answer those questions for you as well.

 

[mathman]

Also, I am wondering why f(x) = mod x is not differentiable at the origin. Surely, f(x) can have two tangents with opposite orientations at the origin, can't it??

Yes it can, but the definition of differentiability requires them to be the same.

 

[Mark44]

By f(x) = mod x, I assume you mean the absolute value function, |x|.
If x < 0, f'(x) = -1
If x > 0. f'(x) = 1
At x = 0, f'(x) does not exist.

For each number c in the domain of f', f'(c) must be a single number, and this goes back to the definition of the derivative as the limit of the difference quotient.

 

[hasan_researc]

Mark44 and mathman:

Why is differentiability defined in that particular way? Would it have mattered if the definition allowed for two or more values of the derivative?

Gib Z : "we want to be able to go on and say that "So and so is Differentiable/Continuous, and because they also satisfy something else, we know that they are also Analytic or something". " Why would you want to do that?

I have stduied limits, but in the physical sciences, so I don't a mathematician's view of limits. But my question is that continuity implies the continuity of a sequence of values (that define the curve) at both ends. But a function is continuous if f(x) tends to f(a) as x tends to a. Should we not also want to include in the def. that f(x) should continue in both dirn. ? I'm not sure whether my lack of a rigorous undetstanding of limits is causing prob here.

 

[Gib Z]

Ok well then it should be clear from the definition of the derivative why |x| has no derivative at x=0.

"But my question is that continuity implies the continuity of a sequence of values (that define the curve) at both ends."

Sorry I'm a bit unsure what you mean by that?

"But a function is continuous if f(x) tends to f(a) as x tends to a. Should we not also want to include in the def. that f(x) should continue in both dirn. ?"

Indeed, for the limit of of f(x) to tend to f(a), we REQUIRE the the function to continue some distance $\delta$ in both directions (left and right), and further more, that for all x values within this range $(a-\delta, a+ \delta)$ we require the value of f(x) to be within some predetermined distance of f(a), $\epsilon$. For the limit to exist, we have to be able to find a suitable $\delta>0$ for every chosen $\epsilon > 0$ so that the above conditions are fulfilled.

 

[Mark44]

Why is differentiability defined in that particular way? Would it have mattered if the definition allowed for two or more values of the derivative?

The derivative f' of some function f is itself a function, and by the definition of a function, for each valid input value (value in the domain of the function), there is a single function value (output value). If someone had defined the derivative in such a way that there were two or more output values, then the derivative would not be a function.

 

[hasan_researc]

The derivative f' of some function f is itself a function, and by the definition of a function, for each valid input value (value in the domain of the function), there is a single function value (output value). If someone had defined the derivative in such a way that there were two or more output values, then the derivative would not be a function.

And why exactly do we want the derivative to be a function?

 

[mathman]

And why exactly do we want the derivative to be a function?

The best answer I can give is that mathematics requires precision in its definitions, theorems, etc. The derivative is defined to be a function, whether or not you like it.

 

[Hurkyl]

Why is differentiability defined in that particular way? Would it have mattered if the definition allowed for two or more values of the derivative?

No. But in a hypothetical world where the derivative was defined in a way that allowed multi-valued functions, we still would have named the special case where the derivative turned out to be single-valued. And in classes we wouldn't teach derivatives, but instead teach the special single-valued case. Entire generations will never hear the word derivative, because the term would only ever come up in the rare situations where the special single-valued case doesn't apply but the derivative tells us something useful, and it is inconvenient to just treat it in an ad hoc fashion.

(Incidentally, see the term "tangent cone")

 

[hasan_researc]

No. But in a hypothetical world where the derivative was defined in a way that allowed multi-valued functions, we still would have named the special case where the derivative turned out to be single-valued. And in classes we wouldn't teach derivatives, but instead teach the special single-valued case. Entire generations will never hear the word derivative, because the term would only ever come up in the rare situations where the special single-valued case doesn't apply but the derivative tells us something useful, and it is inconvenient to just treat it in an ad hoc fashion.

(Incidentally, see the term "tangent cone")

And why would people want to teach the special single-valued case? I mean why has the derivative been defined in the first place?

 

[Gib Z]

Because they are useful in countless problems in mathematics, physics and other sciences. We want it single valued simply for convenience- otherwise we would have to say every single time, out of multiple values, which value we mean for each problem.

 

[hasan_researc]

Derivatives are useful in mechanics, I know. For instance, acceleration is the derivative of velocity. I can't think of acceleration taking two different values and us having to choose one of these. I mean are there any examples where the derivative could have taken two or more values, and we would have had to choose one value only?

Why would we need to choose one value anyway? Why not choose two or more?

 

[NaturePaper]

We'll Mathematicians like to be very precise to make sure everything they "prove" it actually proven and not just probably true. That's why in the last couple centuries it has focused more on rigor than before, and Clearly stating precise definitions is part of that. We need to know exactly what something is, before we can go and say other things have that property.

Well said...Really I am feeling jealous to you for such a nice answer!

Regards,
NP

 

[Hurkyl]

We'll Mathematicians like to be very precise to make sure everything they "prove" it actually proven and not just probably true.

There's a reverse direction too -- we might consider a statement and decide it ought to be the conclusion of a theorem, so we work backwards to figure out what hypotheses will ensure it. (or what adjustments we need to make to the desired conclusion)

 

[AlbertEinstein]

Hi,

I don't understand why mathematicians would need to define the mathematical concepts of diffferentiabilty and conitnuity. To be honest, I don't even understand why "f(x) tends to f(a) as x tends to a" describes continuity.

Also, I am wondering why f(x) = mod x is not differentiable at the origin. Surely, f(x) can have two tangents with opposite orientations at the origin, can't it??

Please help!

Earlier when calculus was invented by Newton and leibniz, then such questions of rigor was absent, or they did not have the right tools for the precise definitions of continuity, until weierstrass. But then as mathematicians looked carefully at these concepts, they were not satisfied with just an informal tone in the language, and they began inventing examples which contradicted common sense and physical intuition. So they had to formulate these concepts in more precise language of epsilon delta, etc.

In physics the type of functions we usually work with are usually well-behaved, and don't pose problem. And they can survive without knowing rigorous mathematics. Now if u wish you can say that the modulus function has left slope as -1, and right slope as 1, but since these numbers are not equal at origin we say that it is not differentiable at origin. The notion of function is the central concept of mathematics and it demands them to be single-valued, simply to remove ambiguity. If you are aware with complex analysis the u may have noticed that analyticity is defined for "functions", and therefore we choose a branch of a "multivalued function" simply to keep things straight.

Coming to the question - "why "f(x) tends to f(a) as x tends to a" describes continuity." i think u can manage it. Give a deep thought or else refer to some gud calc books like spivak or thomas-finney