Continuity of a function in an interval:
Continuity in an open interval:
A function f(x) is called be continuous in an open interval (a, b) if it is continuous at every point of the interval (a, b).
Continuity in a closed interval:
A function f(x) is called be continuous in a enclosed interval [a, b] if
(i) f(x) is continuous at every point of the interval (a, b).
(ii) f(x) is continuous from right at x = a i.e. 
= f(a).
(iii) f(x) is continuous from left at x = b i.e.
= f(b) and;
Geometrical meaning : The function f(x) may be continuous in the enclosed interval [a, b] if the graph of y = f(x) is an not broken line as curved or straight from the point (a, f(a)) to (b, f(b)).
Suppose f(x) and g(x) be two functions, then the subsequent results holds true.
Case I: If f and g both are continuous at x=a, then c1f(x) ± c2(x) and f(x) . g(x) may be continuous at x = a. And f(x)/g(x) may also be continuous at x = a, given g(a) ≠ 0.
Case II: When one of the function f or g is irregular at x=a, then c1f(x) ± c2(x) is definitely discontinuous, but nothing may be called about the continuity of f(x) . g(x) and f(x)/g(x). They can or cannot be continuous at x=a.
Case III: When f and g each are discontinuous at x=a, then nothing may be called about the continuity of c1f(x) ± c2(x f(x) . g(x) and f(x)/g(x).
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