Continuity of a function at a number:
A function f(x) is said to be continuous at x = a, if it satisfies the three conditions:
(i) f(x) is defined at x = a, i.e., f(a) exists
Continuity of a function in an open interval:
A function f(x) is said to be continuous in an open interval (a, b), if it is continuous at every point in (a, b).
Continuity of a function in a closed interval:
A function f(x) is said to be continuous in the closed interval [a, b], if it is continuous at every point of the open interval (a, b) and if it is continuous at the point 'a' from the right and continuous at 'b' from the left.
Ex: Show that f(x) = [x] is right continuous but left continuous at x = 5.
Sol: If x > 5 then [x] = 5
f(x) = [x] = 5 = f(5)
∴ f is right continuous at 5.
If x < 5, then [x] = 4
f(x) = [x] = 4 ≠ f(5)
∴ f is left continuous at 5.
|i. If 'f' and 'g' are real valued continuous functions defined on a neighborhood of point 'c', then f + g, f – g, fg and (provided that g(c) ≠ 0) are also continuous at x = c.
|ii. A polynomial function is continuous on R.
|iii. Any rational function is continuous except at the point where the denominator vanishes.
Intermediate value theorem:
If a function 'f' is continuous on a closed interval [a, b] and 'n' is a number with f(a) ≤ n ≤ f(b),
then there exists a number p in [a, b] such that f(p) = n.