• A function ‘f’ is said to be continuous at x = a iff ‘f’ is defined in some neighborhood of ‘a’ and limit of f(x) when x tends to ‘a’ is equal to f(a).
• Let f be real valued continuous function defined in a neighborhood of a point ‘a’. Then
- kf is continuous at ‘a’ for all k ∈ R.
- k ± f is continuous at ‘a’ for all k ∈ R.
• The set of all points where the function is continuous, is called its domain of continuity.
• A function ‘f’ is said to be continuous in an open interval (a, b) if it is continuous at each point (say c) in a < c < b.
• A function is said to be continuous in the closed interval [a, b], iff
- it is continuous in the open interval (a, b)
- it is right-continuous at ‘a’,
- it is left continuous at ‘b’.
• The function which is not continuous at any point ‘c’ is said to be discontinuous at that point.
• Let f and g be real valued functions continuous at a real number ‘c’. Then
- f ± g is continuous at ‘a’.
- f.g is continuous at ‘a’.
- f/g is continuous at ‘a’, provided g(a) ≠ 0.
• Let ‘f’ and ‘g’ be real functions, such that ‘fog’ is defined at ‘a’. If ‘g’ is continuous at a point ‘a’ and if ‘f’ is continuous at g(a), then (fog) is continuous at ‘a’.
Keywords: Continuous Function, Discontinuous Function, Removable Discontinuity, Algebra of Continuous Functions
Check that function f(x) = 13x2 – 5 is continuous at x = 0 or not.Marks:1
Examine the following function for continuity.
Examine the following function for continuity. f(x) = x – 5.Marks:1
Examine the continuity of the function
F(x) = 3x2 – 2 at x=2.Marks:2
Check that function f(x) = 3x – 5 is continuous at x = 0 or not.Marks:1