Question

defination of continuity

Answer 1

The values of a function f(x) at points near a are good predictors of the value of f at a.

The graph of f is a connected curve with no jumps, gaps, or holes.

Answer 2

Definition: (continuity at a point)
If f(x) is defined on an open interval containing c, then f(x) is said to be continuous at c if and only if
.
source: http://en.wikibooks.org/wiki/Calculus/Continuity

Answer 3

google it and find an explanation you like
it may be much clearer than all the opinions you are about to get

Answer 4

a real-valued function f of a real variable is continuous at a point a precisely if the difference f(a + δ) − f(a) is infinitesimal whenever δ is infinitesimal.

Answer 5

the preimage of any open set is open