Continuity of a Function:
The word continuous means without any gap or break. Generally speaking, a function ƒ is continuous at a point c if ƒ does not have a gap or break in its value at c viz. ƒ(c) when related to the values at points near c. In another words, ƒ is continuous at a point c if as x lies very nearer to c, ƒ(x) is also lies very nearer to ƒ(c) i.e. if |x - c| is build smaller and smaller, | ƒ(x) - ƒ(c)| may also be create smaller and smaller. Suppose a function
ƒ(x) = x2 if 0 ≤ x < 1
= 5 if x = 1
= x2 if 1 < x ≤ 2
It can be seen that in the domain [0, 2], [x - 1] can be as small as we please, yet
|ƒ(x) - ƒ(1)| = |ƒ(x) - 5| ≥ 1 for x ≠ 1
Therefore the function is not continuous at x = 1. However, if we take
ƒ(x) = x2 if 0 ≤ x < 1
= 1 if x = 1
= x2 if 1 < x ≤ 2
Then as |x - 1| is create smaller and smaller.
|ƒ(x) - ƒ(1)| = |ƒ(x) - 1|
may also be made smaller and smaller. Thus ƒ is continuous at x = 1.