Calculus

Continuity

A function f(x) is said to be continuous at x=c if

1. f(c) is defined.
2.  is exist.
3. 

A function is said to be continuous on the interval [a, b] if it is continuous at each point in the interval.

In general, 

Let f(x) be a function defined on R. f(x) is said to be continuous at x = a if and only if

This is equivalent to the definition: A function f(x) is continuous at x = a, if and only if

(a) f(x) is well-defined at x = a, i.e. f(a) exists and f(a) is a finite value,

(b)   exists and .

Sometimes, the second condition may be written as 

A function is discontinuous at x = a if it is not continuous at that point a. There are four kinds of discontinuity:

1. Removable discontinuity:

 exists but not equal to f(a)

1. Jump discontinuity:

1. Infinite discontinuity:

i.e. limit does not exist.

1. At least one of the one-side limit does not exist.

Properties of Continuous Functions

• If f(x) and g(x) are two functions continuous at x = a, then so are f(x) ± g(x), f(x)g(x)
  and  provided g(a) ≠ 0.
• If f(x) is continuous on [a, b], then f(x) is bounded on [a, b]. But f(x) is continuous on (a, b) cannot implies that f(x) is bounded on (a, b).

• If f(x) is continuous on [a, b], then it will attain an absolute maximum and absolute minimum on [a, b].
• If f(x) is continuous on [a, b], there exist x₁ ∈ [a, b] such that f(x₁) ≥ f(x) and x₂ ∈ [a, b] such that f(x₂) ≤ f(x) for all x ∈ [a, b]. x₁ is called absolute maximum of the function and x₂ is called the absolute minimum of the function.

• If f(x) is continuous on [a, b] and f(a)f(b) < 0, then there exists c ∈ [a, b] such that f(c) = 0

Intermediate Value Theorem

• If f(x) is continuous on [a, b], then for any real number m lying between f(a) and f(b), there corresponds a number c ∈ [a, b] such that f(c) = m.

Let f(a) = c and f(b) = d, if f is continuous and strictly increasing on [a, b], then f^-1 is also continuous and strictly increasing on [c, d] (or [d, c]).