check definition of continuity of a function

[Maddy_Math]

Hey guys is following definition of continuity of a function true?

------------------------------------------------------------------------
Def: A function is said to be continuous at "c" if the following are met

i. f(c) is defined
ii. limit f(h+c) as h-->0 exist
iii. limit f(h+c) as h-->0 = f(c)
------------------------------------------------------------------------
if you think it's true then please tell me how cause all I know is, A function is continuous if:

limit f(x) as x-->a = f(a) and obviously this can be met only if both sides of this equation exist but how is that first definition true that doesn't make sense to me

also please if anyone knows how to write math equations in post then tell me.

 

[Quaid]

Hey guys is following definition of continuity of a function true?

------------------------------------------------------------------------
Def: A function is said to be continuous at "c" if the following are met

i. f(c) is defined
ii. limit f(h+c) as h-->0 exist
iii. limit f(h+c) as h-->0 = f(c)
------------------------------------------------------------------------

if you think it's true then please tell me how cause all I know is, A function is continuous at a if:

limit f(x) as x-->a = f(a) and obviously this can be met only if both sides of this equation exist but how is that first definition true that doesn't make sense to me

The definitions are the same, but the second form is more commonly used.

Symbol h represents the distance between variable x and constant c, on the x-axis (i.e., Real number line).

h = x - c

They simply replaced symbol x in f(x) with the expression c+h

As x approaches c, the distance h between them approaches zero. So h-->0 is just a different way of saying x approaches c.

If symbols a and c represent the same number (in the domain of some continuous function f), then we can rewrite either definition to match the other.

\(\displaystyle \lim\limits_{h \to 0} f(c+h) = \lim\limits_{x \to c} f(x)\)

If you have a calculus text, I would expect that the author uses both methods (most do). Let us know, if you're still puzzled over any of this.

Oh, you're right about the last line incorporating the first two lines. That is, if we know that line iii is true, then lines i and ii go without saying. (Some authors list each condition separately, for emphasis I suppose.)

also please if anyone knows how to write math equations in post then tell me.

The math-formatting system is called LaTex. You may google for tutorials, but be warned that there are different implementations of LaTex on the Internet; not everything that you find works here.

You may right-click any LaTex expression (try it above), to see the coding (Show Math As >> TeX commands).

In these forums, you must enclose LaTex coding within [ֺtex] and [/ֺtex] tags.

Ciao :smile:

 

[Maddy_Math]

The definitions are the same, but the second form is more commonly used.

Symbol h represents the distance.........................................

Ciao :smile:

Thankyou Very Much Quaid, that really helped.

yes I missed continuous at "a" in 2nd definition, thanks for correcting

and I were being confused because of the use of two different symbols for constant being "a" and "c" " a = c " resolves the mystery, thanks again

and now let me try if I could LaTex or not

thanks already if it worked

\(\displaystyle \lim\limits{x \to 0} f(x)\)

 

[Maddy_Math]

lolx not that bad let me try again

\(\displaystyle \lim\limits_{x \to 0} f(x)\)

 

[Maddy_Math]

lolx not that bad let me try again

\(\displaystyle \lim\limits_{x \to 0} f(x)\)

lolx... yes that worked Thanks again Quaid

 

[pka]

Hey guys is following definition of continuity of a function true?
------------------------------------------------------------------------
Def: A function is said to be continuous at "c" if the following are met
i. f(c) is defined
ii. limit f(h+c) as h-->0 exist
iii. limit f(h+c) as h-->0 = f(c)
-----------------------------------------------------------------------

Can you find an example such \(\displaystyle \displaystyle{\lim _{x \to c}}f(x) = L\) BUT neither i) nor iii) is true.

 

[Maddy_Math]

Can you find an example such \(\displaystyle \displaystyle{\lim _{x \to c}}f(x) = L\) BUT neither i) nor iii) is true.

yup there are many functions which are not defined at some point but their limit at that point exists like \(\displaystyle \frac{x^2 - 1}{x-1} \text{ here f(1) is not defined but} \lim \limits_{x \to 1} \frac{x^2 - 1}{x-1} \; = \; 2 \)

i. \(\displaystyle \lim \limits_{x \to 1} \frac{x^2 - 1}{x - 1} \) exists
ii. but f(1) does not exist
iii. statement ii follow \(\displaystyle \lim \limits_{x \to 1} \frac{x^2 - 1}{x - 1} \neq \text{f(1)} \\ \TeX \)