Continuity & Uniform Continuity

In summary, continuity is determined by the existence of a d value in relation to e, while uniform continuity has a d value that is independent of x. The function must be continuous at all points in a compact set for it to be uniformly continuous. The definition of continuity provided is correct, while the definition of uniform continuity is incomplete and needs to specify that d is independent of x.
  • #1
mtvateallmybrains
3
0
I'm seeking a bit of affirmation or correction here before i try to solidify this to memory...

I know continuity to mean:
Let f:D -> R (D being an interval we know to be the domain, D)
Let x_0 be a member of the domain, D.
This implies that the function f is continuous at the point x_0 iff
for any e >0 there exists a d>0 such that x belongs to the domain, D AND |x-x_0|< d => |f(x)-f(x_0)| < e .

I interpret this to mean:
This is the criterion by which we judge if some function (f) is continuous at whatever-point-we-wish-to-test-for-continuity-at (x_0) over some interval that is, in the least, a subset of the domain (if not the entire domain itself).

////////

I know uniform continuity to mean:
Let a compact set, K be a subset of R. Let f:K->R. Then f is uniformly continuous on the set K.

I interpret this to mean:
The previous definition of continuity is now applicable to any and every point that is a member of the compact set, K. In other words, the interval/set over which K is defined satisfies the previous criterion of continuity at all points in K.

====
Is there a need to adjust either my definition (as quoted by my prof. for an introductory advanced calculus class) or my interpretation of these concepts - or are they within a reasonable tolerance of "precise-ness" for the _actual_ definition/interpretation/distinction of the concept of continuity and of the concept of uniform continuity? Please advise, thank you!
 
Physics news on Phys.org
  • #2
A simple way to look at it is in the definition of d. In the definition of continuity, d may depend on x. For uniform continuity, d is independent of x. In both cases, d will depend on e.
 
  • #3
so in the case of continuity we select x, and in the case of uniform continuity we select delta in terms of epsilon?
========================
Also, it would probably be most helpful (at least to myself, if not others) if someone were to respond to the first post as if it were four True/False statements: (just like the old days)
IF a statement is True, THEN please say that "Yes, that statement (in it's entirety) is necessarily and sufficiently True." ELSE, the statement is FALSE.
If the statement is False, please explain why (what necessary and sufficient conditions - while also taking into considering the introductory nature/level of this material - were wrong and/or missing & what, if anything, is extraneous?).

This will hopefully eliminate a great deal of confusion and the potential for ambiguity. Also note that I'm not asking for an explanation here (unless what I have asserted is FALSE), I'm asking for an affirmation.

Thanks!
 
Last edited:
  • #4
Right:
We say that a function is continuous on an interval [itex]D[/itex] if for any [itex]x_0[/itex] on the interval, and for any [itex]\epsilon > 0 [/itex] there exists [tex]\delta > 0[/itex] with the property that [itex]x \in D[/itex] and [itex] | x_0 - x| < \delta[/itex] implies that [itex]|f(x)-f(x_0)| < \epsilon[/itex]

A function is unformly continuous on an interval [itex]D[/itex] if for any [itex]\epsilon>0[/itex] there exists [itex]\delta > 0[/itex] so that for any [tex]x_1,x_2 \in D[/tex], [itex]|x_1-x_2|<\delta \Rightarrow |f(x_1)-f(x_2)| < \epsilon[/itex]

For example [tex]f(x)=\frac{1}{x}[/tex] is contiuous, but not uniformly continuous on the interval [tex](0,+\infty)[/tex].
 
Last edited:
  • #5
You don't actually define uniform continuity, so it's hard to say if you are correct or not in that.

A function that is continuous on a compact set is uniformly continuous, yes, but that isn't the definition.

Basically, a function is continuous at a point x in the domain if... etc ... *where d depends on both e and x*, so your definition is correct (I'm not at all sure what your interpretation is an interpretation of, though).

it is uniformly continuous if d can be chosen such that there is no dependence on x.
 
  • #6
To NateTG: In your definition of uniform continuity, you switched epsilon and delta in the final implication.
 
  • #7
mathman said:
To NateTG: In your definition of uniform continuity, you switched epsilon and delta in the final implication.

Oops. (fixing)
 

What is continuity?

Continuity is a mathematical concept that describes the smoothness and connectedness of a function. It means that a function's output values change gradually and consistently as its input values change.

How is continuity different from differentiability?

Continuity and differentiability are related concepts, but they are not the same. A function is continuous if its output values change continuously as its input values change, while a function is differentiable if it has a derivative at every point on its domain.

What is uniform continuity?

Uniform continuity is a stronger version of continuity that ensures a function's output values do not change too much when its input values change slightly. It means that for any small change in the input, there is a corresponding small change in the output.

How is uniform continuity different from continuity?

Uniform continuity differs from continuity in that it requires the small change in the input to correspond to a small change in the output for the entire domain of the function, not just at a specific point. This guarantees a smoother and more predictable behavior for the function.

Why is continuity important in mathematics and science?

Continuity is essential in mathematics and science because it allows us to model and understand the behavior of natural phenomena and real-world systems. Many physical laws and mathematical models rely on the assumption of continuity to make accurate predictions and describe the behavior of complex systems.

Similar threads

Replies
4
Views
834
Replies
2
Views
1K
Replies
18
Views
2K
Replies
14
Views
1K
Replies
3
Views
2K
Replies
3
Views
1K
  • Calculus
Replies
13
Views
1K
  • Calculus
Replies
13
Views
2K
Back
Top