Continuous Functions

Let f be a function defined on an interval I = [a, b]. A continuous function on I is a function whose graph y = f(x) can be described by the motion of a particle travelling along it from the point (a, f(a)) to the point (b, f(b)) without moving off the curve.

Continuity at a Point

If the left hand limit, right hand limit and the value of the function at x = c exist and equal to each other, then f is said to be continuous at x = c.

Continuity in an Interval

A function f is said to be continuous in an interval [a, b] if it is continuous at each and every point of the interval.

Discontinuous Functions

A function f is said to be discontinuous at a point c of its domain if it is not continuous at c. The point c is then called a point of discontinuity of the function.

Algebra of Continuous Functions

Theorem 1

If f and g be two real functions continuous at a real number c. Then

  • f + g is continuous at x = c
  • f – g is continuous at x = c
  • f . g is continuous at x = c
  • f/g is continuous at x = c

Theorem 2

If f and g are real valued functions such that (fog) is defined at c. If g is continuous at c and if f is continuous at g(c), then (fog) is continuous at c.