what is the meaning of "continuum set"? Can you explain and give an example?

Answer 1

Selenga,

I am afraid that the above answers are misleading. Plenty of infinite sets do not form a continuum (eg the integers) so the infinitude of the real numbers is not the important issue. Also, it has nothing to do with continuity of functions in the way that the second answerer implied.

The idea behind a continuum set is that it is a set of numbers with no gaps or holes. Strictly, you need to conditions:

1 between any two distinct numbers in the set there is another number in the set

2 any (non-empty) bounded subset has a least upper bound in the set.

So, for example, the rational numbers (ie the fractions) satisfy the first condition but not the second because the bounded set, (1, 1.4, 1.41, 1.414, 1.4142, ...) does not have an upper bound in the set of rational numbers. (Note that each member of the set IS rational 1/1, 14/10, 141/100, etc but that the limit, the square root of 2, is NOT rational and so there is no smallest number in the set that is bigger than every member of the subset).

This is all quite subtle but what it really means is that in a continuum there is an infinitude of numbers and no gaps anywhere.

I hope that this makes sense and helps. But PLEASE don't be taken in by misleading (but apparently simpler) answers. It's quite a subtle concept, I'm afraid!


Perspy

Answer 2

The members of the set form a continuum, so there are an infinite amount. Example: the set of real numbers between 10 and 100.

Answer 3

A continuous line is whole all the distance.
A continuous function has a value for all numbers (all the numbers on the x-axis for example).

The function 1/x is discontinuous in 0(zero).
Tan x is discontinuous in 90 degree.

I hope this examples explain a little for you.