What are the requirements for a function to be continuous at a point?

[rbzima]

Hey guys, I'm in a little bit of a jam here:
I managed to miss a really important lecture on continuity the other day, and there were a few examples that the professor provided to the class that I just got, but would love it if someone could explain them to me.

First, f(x)=x³ is continuous on all of R. Everyone went through this, but I'm not really sure how it works.

Second, we used the fact that "a function $$f:A \rightarrow$$ R fails to be uniformly continuous on A if and only if there exists a particular $$\epsilon_{0} > 0$$ and two sequences (x_n) and (y_n) in A satisfying

$$\left| x_n - y_n\right| \rightarrow 0$$ but $$\left| f(x_n) - f(y_n)\right| \rightarrow \epsilon_0$$

to show fis not uniformly continuous on R.

Thirdly, we showed that f is uniformly continuous on any bounded subset of R

 

[mdnazmulh]

I studied continuity in the last semester. I can't remember all the theorems ATM.
Any way, as far as I can remember that a function, in plain words, can be continuous
1) at a point or,
2) can be continuous on a certain interval.

For continuity at a point it must have a value at that point
For an interval, it must be continuous at every points in the interval.

I can remember upto this.

 

[soul]

Hey guys, I'm in a little bit of a jam here:
I managed to miss a really important lecture on continuity the other day, and there were a few examples that the professor provided to the class that I just got, but would love it if someone could explain them to me.

First, f(x)=x³ is continuous on all of R. Everyone went through this, but I'm not really sure how it works.

Second, we used the fact that "a function $$f:A \rightarrow$$ R fails to be uniformly continuous on A if and only if there exists a particular $$\epsilon_{0} > 0$$ and two sequences (x_n) and (y_n) in A satisfying

$$\left| x_n - y_n\right| \rightarrow 0$$ but $$\left| f(x_n) - f(y_n)\right| \rightarrow \epsilon_0$$

to show fis not uniformly continuous on R.

Thirdly, we showed that f is uniformly continuous on any bounded subset of R

For the first quesiton, you can use the contunuity of limit. As I remember ıt is something like that. lim(x --> a) f(x) = f(a). and for f(x^3), this statement is correct.

 

[Diffy]

I studied continuity in the last semester. I can't remember all the theorems ATM.
Any way, as far as I can remember that a function, in plain words, can be continuous
1) at a point or,
2) can be continuous on a certain interval.

For continuity at a point it must have a value at that point
For an interval, it must be continuous at every points in the interval.

I can remember upto this.

This is incorrect.

One counter example is the following function

Let $$f(x)=\left\{\begin{array}{cc}0,&\mbox{ if } x\neq 0\\1, & \mbox{ if } x=0\end{array}\right.$$

This function has a value at x = 0, but is certainly not continuous at that point.

From Wiki,
To be more precise, we say that the function f is continuous at some point c when the following two requirements are satisfied:

f(c) must be defined (i.e. c must be an element of the domain of f).

The limit of f(x) as x approaches c must exist and be equal to f(c).