 # Continuity

## •    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

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• Q1

Check that function f(x) = 13x2 – 5 is continuous at x = 0 or not.

Marks:1 • Q2

Examine the following function for continuity. Marks:1 • Q3

Examine the following function for continuity. f(x) = x – 5.

Marks:1 • Q4

Examine the continuity of the function
F(x) = 3x2 – 2 at x=2.

Marks:2  