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Continuity: Examples, Theorems, Properties and Notes



We can say a function f ( x ) to be continuous at x = a if \(\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\). In this section we will read continuity equations and functions.

Also, We can say a function to be continuous on the interval [a , b] if it is continuous at each and every point in the interval.

Continuous function:
Graphically we can say that if the graph of the function has no gap only than it is continuous function.

Discontinuous function:
A function, the graph of which has gaps or that function is not continuous is discontinuous function.

Example: See the graph of continuous function:
continuity of function

Also, the graph of discontinuous function:

discontinuous continuity

Continuity of a function at a point:

For a real function f within its domain a. We can only say f is continuous at a if \(\lim \limits_{x \to a}f(x)\) = f(a)
we can simply say it as \(\lim \limits_{x \to a^{-}}f(x) = \lim \limits_{x \to a^{+}}f(x) = f(a)\).

At this point a function f(x) is continuous, if \(\lim \limits_{x \to a^{+}}f(x)\) exists and is exactly equal to f(a).

Discontinuity of a function:

Some of the criteria are there for the discontinuity of function are:

  1. F(a) is not defined.
  2. When left hand limit (LHL) and right hand limit (RHL) both exist but is unequal.
    e. \(\lim \limits_{x \to a^{-}}f(x) ≠ \lim \limits_{x \to a^{+}}f(x)\)
  3. Either \(\lim \limits_{x \to a^{-}}f(x)\) is infinite (or doesn’t exist) or \(\lim \limits_{x \to a^{+}}f(x)\) is infinite (or doesn’t exist).
  4. LHL and RHL both exists but is not equal to F(a)
    e. \(\lim \limits_{x \to a^{-}}f(x) = \lim \limits_{x \to a^{+}}f(x)\) ≠ f(a).

Key points:

  1. If F(x) is continuous and g(x) is discontinuous at x = a, then we can’t say that the product function is discontinuous. i.e. Ø(x) = f(x).g(x) is not necessarily be discontinuous at x= a.
  2. If both f(x) and g(x) are discontinuous at x=a, then the product function Ø(x) = f(x).g(x) may be or may not be or discontinuous.

Continuity of function in an interval:

  1. A function f(x) will only be continuous in (a, b) (open interval) if f(x) is continuous at each and every point in that interval.
  2. A function f(x) will only be continuous in [a, b] (closed interval) if f(x) is continuous at each and every point in that interval. The fact to be considered in it is that f(x) must be continuous at left end point x = a and also on right-hand side end point x = b.

Fundamental theorems of continuity:

  1. If f and g are both continuous functions, then
  • f + g, f – g, and fg are continuous function.
  • f is also continuous, where k is constant.
  • \(\frac{f}{g}\) is continuous only at that point where g(x) ≠ 0.
  1. If g(x) is continuous at point “a” and f(x) is continuous at point g(a) then function “fog” must be continuous at “a”.
  2. The best thing about continuity in a closed interval [a, b], If f(x) is continuous in [a,b] then it must be bounded in [a, b].
    we can say that there is some minimum value of f(x) i.e. r and having maximum value R
    As, r ≤ f(x) ≤ R, ɏ x ∈ [a, b].
  3. If f is continuous in [a, b], then corresponding values of minimum and maximum are assumed as values of f(x). Thus,
    a ≤ x ≤ b ⇒ m ≤ f(x) ≤ M or range of f(x) = [m, M], x ε [a, b].
  4. f (x) would have the same sign as f(a) only if f is continuous at a and f (a) ≠ 0, then there exists an interval (a — δ, a + δ) such that for all x ε (a — δ, a + δ).
  5. It is obvious for a continuous function f defined on [a, b] such that f (a) and f (b) are of opposite sign, then there exists at least one solution the equation f(x)= 0 in the open interval (a, b).
  6. If f is continuous in its domain, then |f| also must be continuous in it’s domain.
  7. If f is continuous on [a, b] and this function maps [a, b] into [a, b], then for some x ε [a , b], the the vale obtained is f (x)= x.
  8. When we talk of continuity of f in its domain, then it’s a true fact that 1/f is continuous at Domain – {x : f(x)= 0}
  9. A function f(x) is continuous throughout if it is continuous on the entire real line i.e. (-∞, ∞).

Intermediate value theorem (IMVT):

For studying continuity in an interval we study this IMVT. Let’s think that f ( x ) is continuous on [ a , b ] and let M be any number between f (a) and f (b). Then there exists a number c such that,

  • a < c < b
  • f(c)=M
    intermediate mean value theorem continuity

 Continuity Equation Examples:

  1. Check the continuity of \(\frac{5x^{2}+2}{x – 4}\).
    At x=4, the value of denominator is 0.

    So the function is discontinuous at x=4.
  2. Solve the limit as given \(\mathop {\lim }\limits_{x \to 0} {{\bf{e}}^{\sin x}}\).
    You must be aware that exponentials are continuous everywhere we can use this fact to solve this problem.

    \(\mathop {\lim }\limits_{x \to 0} {{\bf{e}}^{\sin x}} = {{\bf{e}}^{\mathop {\lim }\limits_{x \to 0} \sin x}} = {{\bf{e}}^0} = 1\)

    More from Calculus
    Relation and Functions Limits Formula
    Differentiability Rules Derivative Formula
    Integral Formula Inverse Trigonometric function
    Application of Integrals Logarithm Formulas
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