Continuity of functions | Learn Maths Online

Continuity of functions

The word continuous means without any break or gap. Continuity of functions exists when our function is without any break or gap or jump. If there is any gap in the graph, the function is said to be discontinuous.

Graph of functions like sinx, cosx, secx, 1/x etc are continuous (without any gap) while the greatest integer function has a break at every point(discontinuous).

1. A function f(x) is said to be continuous at x = c,  if  {\lim }\limits_{x \to c} f(x) = f(c) .

 

symbolically f is continuous at x = c if  {\lim }\limits_{x \to c - h} f(c + h) = {\lim }\limits_{x \to c - h} f(c - h) = f(c).

 

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighborhood of x = a, not necessarily at x = a.

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Reasons for dis Continuity of functions

(i)  {\lim }\limits_{x \to c} f(c)  does not exist   i.e.   {\lim }\limits_{x \to {c^ - }} f(c) \ne {\lim }\limits_{x \to {c^ + }} f(c)

 

(ii)  f(x) is not defined at x= c

 

(iii)   {\lim }\limits_{x \to c} f(x) \ne f(c)

 

Types of Dis Continuity of functions

 Removable type of discontinuities- In this case  {\lim }\limits_{x \to c} f(x)exists but is not equal to f(c) then the function is said to have a removable discontinuity or discontinuity of the first kind. In this case we can redefine the function such that f(x) = f(c) & make it continuous at x= c.

Removable type of discontinuity can be further classified as:

 

(a) Missing Point Discontinuity- Where f(x) exists finitely but f(a) is not defined.e.g.

f(x) = \frac{{(1 - x)(9 - {x^2})}}{{(1 - x)}} here f(x) has a missing point discontinuity at x = 1 , and f(x) = \frac{{\sin x}}{x}

has a missing point discontinuity at x = 0

 

(b) Isolated Point Discontinuity- Where f(x) exists & f(a) also exists but

 {\lim }\limits_{x \to c} f(x) \ne f(c)   e.g.  {\lim }\limits_{x \to c} f(x) = \frac{{{x^2} - 16}}{{x - 4}} here  x \ne 4  & f (4) = 9 has an isolated point

discontinuity at x = 4. Similarly f(x) = [x] + [ –x] = \left[ {_{ - 1....if...x \notin I}^{0....if...x \in I}} \right.  has an isolated point discontinuity at all x \in I.

Also READ this-  IB Maths (Part-1)- Functions-An Introduction to functions in Mathematics

 

Non-Removable type of dis Continuity of functions

In case f(x) does not exist then it is not possible to make the function continuous by redefining it. Such discontinuities are known as non-removable discontinuity or discontinuity of the 2nd kind.

Non-removable type of discontinuity can be further classified as:

(a) Finite discontinuity- e.g. f(x) = x – [x] at all integral,  f(x) = \frac{1}{{{{\tan }^{ - 1}}x}}

f(x) = at x = 0 and     f(x) = \frac{1}{{1 + {2^{\frac{1}{x}}}}}  at x = 0  [note that f(0+) = 0 ; f(0) = 1]

 

(b) Infinite discontinuity-  e.g. f(x) = \frac{1}{{x - 4}}  or  g(x) = \frac{1}{{{{(x - 4)}^2}}}  at x = 4, f(x) = {2^{\tan x}}

at x =0 and  f(x) = \frac{{\cos x}}{x}  x = 0.

(c) Oscillatory discontinuity- e.g.  f(x) = \sin \frac{1}{x}  at x = 0. In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but   {\lim }\limits_{x \to a} f(a)  does not exist.

4.The Jump Of Discontinuity- In case of discontinuity of the second kind the non-negative difference between the value of the RHL at x = c & LHL at x = c is called The Jump Of Discontinuity. A function having a finite number of jumps in a given interval I is called a Piece Wise Continuous or Sectionally Continuous function in this interval.

5. All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains.

6. If f & g are two functions that are continuous at x= c then the functions defined by :

F1(x) = f(x) ± g(x)  ;  F2(x) = K f(x) , K any real number  ; F3(x) = f(x).g(x) are also continuous at x= c. Further, if g (c) is not zero, then  {F_4}\left( x \right){\rm{ }} = \frac{{f(x)}}{{g(x)}}  is also continuous at   x= c.

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Theorems of Continuity 

(a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function\varphi \left( x \right){\rm{ }} = {\rm{ }}f\left( x \right).{\rm{ }}g\left( x \right) is not necessarily be discontinuous at x = a. e.g. f(x) = x &                            

g(x) =\left[ {_{0....if...x = 0}^{\sin \frac{x}{2}....if...x \ne 0}} \right.

 

(b) If f(x) and g(x) both are discontinuous at x = a then the product function \varphi \left( x \right){\rm{ }} = {\rm{ }}f\left( x \right).{\rm{ }}g\left( x \right)

 is not necessarily be discontinuous at x = a. e.g    {\rm{f(x) = - g(x) = }}\left[ {_{ - 1...if...x < 0}^{1....if...x \ge 0}} \right.

 

(c) Point functions are to be treated as discontinuous. eg. f(x) = \sqrt {1 - x} + \sqrt {x - 1}   is not continuous at x = 1.

 

(d) A Continuous function whose domain is closed must have a range also in closed interval.

(e) If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is

continuous at x = c. eg.   f(x) = \frac{{x\sin x}}{{{x^2} + 2}}  &   g(x) = \left| x \right|   are continuous at x = 0 , hence the

composite   (gof) = \left| {\frac{{x\sin x}}{{{x^2} + 2}}} \right|   will also be continuous at x = 0

 

(f) f(x) = {a_0}{x^0} + {a_1}{x^1} + {a_2}{x^2} + {a_3}{x^3} + ............{a_n}{x^n} this nth degree polynomial is continuous for x \in R

 

(g)  y=Sinx, y=Cosx are continuous  for x \in R

 

(h)  y= {\log _a}x is continuous  for all x>0

 

(i)  y= {a^x} is continuous  for all   x \in R

7. Continuity In An Interval

(a) A function f is said to be continuous in (a, b) if f is continuous at each & every point     Î \in (a, b)

(b) A function f is said to be continuous in a closed interval [a,b] if:

(i) f is continuous in the open interval (a, b) &

 

(ii) f is right continuous at ‘a’ i.e.   {\lim }\limits_{x \to {a^ + }} f(x) = f(a) = a finite quantity.

 

(iii) f is left continuous at ‘b’ i.e.    {\lim }\limits_{x \to {b^ - }} f(x) = f(b) = a finite quantity. Note  that a function f which is continuous in possesses the following properties :

 

Note  that a function f which is continuous in [a,b] possesses the following properties:

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(i) If f(a) & f(b) possess opposite signs, then there exists at least one solution of the equation f(x) = 0 in the open interval (a , b).

(ii) If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open interval (a, b).

 

(ii) If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open interval (a, b).

Sandwich Theorem or Squeeze Theorem

Suppose    f\left( x \right) \le g\left( x \right) \le h\left( x \right)   \forall ,x \ne c  in some interval about c and that f(x) and h(x) approaches the same limit L as approaches c i.e

                     {\lim }\limits_{x \to c} f(x) = {\lim }\limits_{x \to c} h(x) = L    then     {\lim }\limits_{x \to c} g(x) = L

This theorem is known as Sandwich Theorem.

Intermediate Value Theorem

 If we have a function f(x) that is continuous in the closed interval [a,b] and we suppose M number between f(a) and f(b) then there exists a number c in such a way that-

(i)    a < c < b

(ii)   f(c)=M

Download the PDF at practice at least one question on each concept so that you remember them forever. In case of any problem take help from Online Maths Tutors for free

 

 

 limits and continuity module.pdf

​​ practice problems limits and continuity.pdf

You can go to the following links to read the posts about limits
Post on limits- part one

Post on limits-part two

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