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