Continuities and Discontinuities of Real Valued Function

Continuity- A function is called continuous at x=a, if

                        (i)  f(x) is defined at x = a, i.e. f(x) = f(a) at x=a.

                       (ii) left and right hand limits exists at x=a.

                      (iii) f(a+) = f(a-) = f(a).

Discontinuity- A function f(x) which is not continuous at x=a, is called discontinuous at x=a.

Continuous at x₀- Let E ⊂ ℝ, f : E → ℝ, if x₀ ∈ E, then f is continuous at x₀ iff for each ε > 0 there exist δ > 0 : if
|x - x₀| < δ, x ∈ E, ⇒ |f(x) - f(x₀)| < ε.

If f is continuous at x for each x ∈ E, then f is continuous (on E).

Uniformly Continuity- A function f : D → ℝ is uniformly continuous on E ⊂ D iff for every ε > 0, there is δ > 0 : if 

x, y ∈ E with |x - y| < δ, then |f(x) - f(y)| < ε. If f  is uniformly continuous on D, f is uniformly continuous.

Closed set- A set E ⊂ ℝ is closed iff every accumulation point of E belongs to E.

Open set- A set A ⊂ ℝ is open iff for each x ∈ A, there is a neighbourhood N_x of x such that N_x ⊂ A.

Compact set- A set E is compact iff for every family 

{G_α}_α∈A of open sets E ⊂ ∪ G_α , α ∈ A, there is a finite set 

{α₁, . . . , α_n} ⊂ A : ⊂ ∪G_αi , i = 1, 2, ... , n.

Right  continuous-  If E ⊂ ℝ, f : E → ℝ, and x₀ ∈ E, the function f is right continuous at x₀ iff for each ε > 0, there is δ > 0 : x₀ ≤ x ≤ x + δ, x ∈ E ⇒ |f(x) - f(x₀)| < ε.

Left  continuous-  If E ⊂ ℝ, f : E → ℝ, and x₀ ∈ E, the function f is left continuous at x₀ iff for each ε > 0, there is 
δ > 0 : x₀ - δ ≤ x ≤ x₀, x ∈ E ⇒ |f(x) - f(x₀)| < ε.

Types of Discontinuities:

• Discontinuty of first kind- A function f(x) has discontinuity of first kind at x=a, if left and right hand limit exists at x=a, but are distinct. f(a+) ≠ f(a-).
• Discontinuity of second kind-A function f(x) has discontinuity of second kind at x=a, if left and right hand limits does not exists. i.e., neither f(a+) nor f(a-) exist.
• Mixed Discontinuity- A function f(x) has mixed discontinuity at x=a, if either of left or right hand limit exist.
• Removal Discontinuity- A function f(x) has the removal discontinuity at x=a, if f(a+) and f(a-) exist but f(a+) = f(a-) ≠ f(a).
• Irremoval Discontinuity- A function f(x) has irromoval discontinuity at x=a, if f has discontinuity of first kind, second kind or mixed discontinuity.
• Jumps and Jump Discontinuity- If f(a+) and f(a-) exist at x=a, then

           1. f(a) - f(a-) is called left hand jump of f at a.

           2. f(a+) - f(a) is called right hand jump of f at a.

           3. f(a+) - f(a-) is called jump of f at a.

           If any of these jumps is different from 0, then a is called the jump discontinuity of f.

           Jump discontinuity are discontinuity of first kind.

• Infinite Discontinuity- A function f(x) has infinite discontinuity at x=a, if any of the four functional limits f(a+), f(a-), suprimum of f(a+), infimum of f(a-) are indefinitely large or infinite.
• Saltus (Measure of discontinuity)- The saltus of a function f(x) at x = a is the greatest positive difference between any two of the five numbers f(a), suprimum of f(a+), infimum of f(a+), suprimum of f(a-), infimum of f(a-).
• Saltus on right- The greatest positive difference between any two of the three f(a), suprimum of f(a+), infimum of f(a+).
• Saltus on left- The greatest positive difference between any two of the three f(a), suprimum of f(a-), infimum of f(a-).
  Imp. Saltus is zero at point of continuity and greater that zero at point of discontinuity.