Why does continuity still feel weird?

[RicoGerogi]

Been working through some advanced calc problems and the definition of continuity keeps feeling kinda weird. The epsilon-delta thing doesn't quite click intuitively for all cases, especially with those piecewise functions. Seems like there's a persistent disconnect between the formal proof and practical visualization...

 

[PeroK]

Been working through some advanced calc problems and the definition of continuity keeps feeling kinda weird. The epsilon-delta thing doesn't quite click intuitively for all cases, especially with those piecewise functions. Seems like there's a persistent disconnect between the formal proof and practical visualization...

An alternative, which is equivalent to the usual epsilon-delta definition is:

A function $f$ is continuous at a point $a$ if for every sequence (in the domain of $f$) that converges to $a$, the sequence of function values converges to $f(a)$.

This is often more useful practically for proving that a function is not continuous at $a$, as all you have to do is find a sequence $x_n \to a$ where $f(x_n)$ does not converge to $f(a)$.

That said, you need to know the epsilon definition of convergence for a sequence.

 

[jedishrfu]

This may not be germane to the subject of this thread, but epsilon-delta proofs have been the bane of many calculus students, myself included, despite having been a physics major.

There is a calculus text by Keisler that introduces hyperreal numbers and infinitesimals early on, postponing the full epsilon-delta treatment until later in the course. Hyperreals provide a rigorous foundation for working with infinitesimals as actual mathematical objects rather than merely intuitive quantities.

One attraction of this approach is that many of the manipulations traditionally used by physicists and engineers become easier to justify. For example, expressions such as
#dz/dy · dy/dx · dx/dt = dz/dt# can be understood in a way that resembles the informal cancellation of infinitesimals, while remaining mathematically rigorous within the hyperreal framework.

CAVEAT: Of course, most modern analysis texts still develop calculus using the standard epsilon-delta approach, and the two frameworks ultimately describe the same underlying mathematics. Nevertheless, for many students, infinitesimals provide a more intuitive entry point into the subject.

https://people.math.wisc.edu/~hkeisler/calc.html

 

[PeroK]

If you have to sit a standard analysis exam, then appealing to hyperreal numbers won't help.

 

[jedishrfu]

No but hyperreals could offer a check on your answer which I realize is a poor second to, getting the epsilon-delta proof correct in an analysis test.

—-

Some time ago, a physicist gave a great way to look at the epsilon-delta definition by changing the names of its component values to error bounds and tolerances.

Suppose you want the function value (y) to be within some specified error tolerance of its target value. The question is:

How closely must the input (x) be controlled to guarantee that level of accuracy?

In epsilon-delta language:

• ε (epsilon) is the allowed error in y.
• δ (delta) is the required bound on x.

So the statement $\lim_{x\to a} f(x)=L$ means:

Given any tolerance around the desired y value, there exists a tolerance around x such that whenever x stays within that bound, y is guaranteed to stay within the requested tolerance.

For example, if $f(x)=2x+1$ and we want y to be within $0.1$ of 7, then:

$|f(x)-7|<0.1.$

This requires that $|x-3|<0.05.$

Thus, to guarantee an output error of at most $0.1$, the input must be controlled within $0.05$ of 3.

In essence, epsilon-delta is a rigorous way of answering the question:

“If I need the output this accurate, how accurate must the input be?”

That’s very close to how engineers and experimental scientists naturally think about error propagation and tolerances.

 

[RicoGerogi]

An alternative, which is equivalent to the usual epsilon-delta (don't confuse epsilon with the reversed 3 symbol, they both are different) definition is:

A function $f$ is continuous at a point $a$ if for every sequence (in the domain of $f$) that converges to $a$, the sequence of function values converges to $f(a)$.

This is often more useful practically for proving that a function is not continuous at $a$, as all you have to do is find a sequence $x_n \to a$ where $f(x_n)$ does not converge to $f(a)$.

That said, you need to know the epsilon definition of convergence for a sequence.

Does the sequential definition also simplify proving continuity or mainly for disproving?

 

[PeroK]

Does the sequential definition also simplify proving continuity or mainly for disproving?

It depends. Generally, it's not any easier for proving a function is continuous - because you have to consider every possible sequence. Often it will be similar to a regular epsilon-delta proof.

It's quite clumsy to write down the epsilon-delta requirement for a function not to be continuous at $a$. You should try it!

Whereas, finding a single sequence often gets to the heart of why a function is not continuous.

It's also a good exercise to show that the sequence definition is equivalent to the regular definition.

 

[RicoGerogi]

That epsilon-delta negation is definitely clumsy, a single sequence feels much cleaner

 

[Mike_bb]

One attraction of this approach is that many of the manipulations traditionally used by physicists and engineers become easier to justify. For example, expressions such as
#dz/dy · dy/dx · dx/dt = dz/dt# can be understood in a way that resembles the informal cancellation of infinitesimals, while remaining mathematically rigorous within the hyperreal framework.

Infinitesimals like $dx=1/N$ (Leibniz's) work in Physics but most of mathematicians refuse them because such infinitesimals don't have rigorous formalization like NSA's infinitesimals.

 

[RicoGerogi]

That's a great point!

 

[mathwonk]

A little symbolic logic notation is useful for easily negating basically anything. I.e. to negate, you exchange the type (universal vs existential) of the quantifiers and then negate the statement.
I.e. the negation of "for all ( ), for some [ ], for all < >, P", is just
"for some ( ), for all [ ], for some < >, not P".

And the negation of "A implies B", is "A is true but B is false", so to negate
"for every epsilon, there is a delta, such that for all x,
|x-a|<delta implies |f(x)-f(a)| < epsilon",

you just mechanically get:
"there is an epsilon, such that for every delta, there is some x with
|x-a| < delta and yet |f(x)-f)a)| ≥ epsilon."

Happily they taught this formalism in my high school, from Principles of Mathematics, by Allendoerfer and Oakley, chapter 1, Logic, (it's still on my shelf).

 

[mathwonk]

In fact, when stated precisely, the sequential version is even more complicated, i.e. it is being abbreviated above by the unexplained phrase “{xn}—>a”. The full definition of that symbol of course is:

“for all epsilon, there is N such that, for all j, j≥N implies |xj-a|< epsilon”.

Hence the precise negation of the sequential version of continuity is:

“There is a sequence {xn} such that, for all epsilon, there is N such that, for all j, j≥N implies |xj-a|< epsilon, and yet for some epsilon, and all N, there is a j such that j≥N and |f(xj)-f(a)| ≥ epsilon.”

Thus to actually use the sequential version, even to check that continuity fails, you also need to grasp the statement and use of the epsilon-delta, or epsilon-N, formalism. It was initially hard I think for all of us, certainly for me, but it does eventually get easier. The great Mike Spivak recommended just memorizing it like a poem. For me, learning the symbolic logic notation was also a big help. And physics based intuition is certainly a plus.

 

[jedishrfu]

Infinitesimals like $dx=1/N$ (Leibniz's) work in Physics but most of mathematicians refuse them because such infinitesimals don't have rigorous formalization like NSA's infinitesimals.

Infintesimals are part of the hyperreals and so basic arithmetic is allowed ie canceling like infintesimals. Hyperreal number field was rigorously defined in mathematics by work from Abraham Robinson in 1961.

 

[MidgetDwarf]

Been working through some advanced calc problems and the definition of continuity keeps feeling kinda weird. The epsilon-delta thing doesn't quite click intuitively for all cases, especially with those piecewise functions. Seems like there's a persistent disconnect between the formal proof and practical visualization...

Rule of thumb:

If you want to prove that something is continuous, use Epsilon delta definition of limit.

If you want to disprove continuity use sequential formulation.

Try this for the following.

Is the function f defined as f(x) = 1 for x=0 and f(x)=0 for x not equal to 1 continously at 0? What about everywhere else besides 0.

Try this, and we can work on more interesting problems.

 

[Dale]

Hyperreals provide a rigorous foundation for working with infinitesimals as actual mathematical objects rather than merely intuitive quantities.

The nice thing about that is that having a rigorous approach to infinitesimals allows you to correct some of the errors that people make while thinking just intuitively about infinitesimals.

 

[mathwonk]

midget dwarf, nice example, but there is a typo in your statement; likely you meant f(0) = 1, but f(x) = 0 for x ≠ 0.

 

[mathwonk]

I apologize for provoking sad faces by post #12. To be honest, most people will not need to give an actual proof of sequential convergence to be convinced it holds, so epsilons, N's, and deltas can be kept in the background.

E.g. in the example of midgetdwarf, we can pretty much assume that most people will agree that the constant sequence xn = 0 for all n, converges to 0, without proving it. And they will also agree that this sequence does not converge to any other number, in particular it does not converge to 1. They will also probably grant that the sequence 1/n, converges to 0, without proof.
Hence we can take as our test sequence xn = 1/n, and then since f(1/n) = 0 for all n, we have that 1/n -->0, and that f(1/n) = 0-->0 ≠ 1 = f(0). Hence f is not continuous, although we did not actually prove it completely from the definitions. I.e. it is not easier to use sequences to actually prove continuity fails, but it is easier to use them to give a convincing argument, even if it omits some details of the proof.