Continuity of functions-IB Maths topics

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 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.

<img src="continuous functions.png" alt="continuous functions">

2.Reasons of discontinuity:

(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)

 

3.Types of Discontinuities

 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.

 

Non-Removable type of discontinuities- 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.

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:

(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

 

Continuity_G2

 

Example 1  Given the graph of f(x), shown below, determine if f(x) is continuous at , and .

Continuity_G1

 

Solution        First x=-2                   

The function value and the limit are different so the function is discontinuous at this point.  This type of discontinuity in a graph is considere as  jump discontinuity. It occurs where the graph has a break in it.

Now at x=0                      

So the function is continuous at this given point.

 

At x=3         

Clearly, the function is discontinuous here. Graph is exihibiting a whole at given point. we call it removable discontinuity

Example-2  Check the continuity of the below function

Ans- Rational function can be discontinuous if its denominator is zero

so the given function is continuous everywhere except x=5 and x=-3

Here I am posting practice problems on ‘limits and continuity’ as well as a module on the same topics

 limits and continuity module.pdf

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 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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