Question

Please explain:--
"A real function f(x) is said to be continuous if it is continuous at every point in the domain of f(x)."

Answer 1

thats how continuity is defined
that means at every point in the domain, limit h->0 f(x+h) = f(x)
the direction in which you approach the point does not matter

Answer 2

I didn't understand my matter.Please tell me in more easier way.

Answer 3

continuity is a "local property" i.e. we say \(f\) is continuous at a point \(a\) if \(\lim_{x\to a}f(x)=f(a)\)

Answer 4

in other words, the definition of continuity is at a number.
therefore to call a function "continuous" means you have to reference both the function and the numbers at which it is continuous

Answer 5

so to say the function itself is continuous means it is continuous on all the numbers in its domain. that is for any \(a\) in the domain of \(f\) we have
\[\lim_{x\to a}f(x)=f(a)\]

Answer 6

& what about range?

Answer 7

it should be continuous at every point in that range

Answer 8

K!

Answer 9

continuity has nothing to do with that
lets make an analogy.
suppose you have a friend you think is tall. that friend is probably tall all the time.

whereas if you think your friend is a thief, he is clearly not stealing every second of the day, but only some times.

so being a thief is a local property.

we don't make that distinction so much in english because you might say that your friend is a thief, even though he only steals sometimes. in math it doesn't work that way. you cannot say "a function is continuous' without saying when (or where)

Answer 10

oh I see! thanx to all.

Answer 11

http://oregonstate.edu/instruct/mth251/cq/Stage4/Lesson/continuity.html
check this link out
you'll get a good idea on continuity.
@satellite73 I think we can tell that a function is continuous with out asking at what point if it is continuous over the entire domain.

Answer 12

K!Thanx & I understood it all the way.