{"id":431,"date":"2017-09-13T00:42:45","date_gmt":"2017-09-12T19:12:45","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=431"},"modified":"2025-05-27T00:47:55","modified_gmt":"2025-05-26T19:17:55","slug":"continuity-of-functions","status":"publish","type":"post","link":"https:\/\/ibelitetutor.com\/blog\/continuity-of-functions\/","title":{"rendered":"Continuity of functions | Learn Maths Online"},"content":{"rendered":"

Here is a detailed discussion about continuity of functions by our IB Maths Tutors<\/span>. But lets first discuss its definition.<\/p>\n

Continuity of functions<\/span><\/h2>\n

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.<\/p>\n

Graph of functions like sinx,\u00a0cosx,\u00a0secx, 1\/x etc are continuous (without any gap) while the greatest integer function has a break at every point(discontinuous).<\/p>\n

1. A function f(x) is said to be continuous at x = c, \u00a0if\u00a0\"\u00a0.<\/p>\n

symbolically\u00a0f is continuous at x = c if\u00a0\".<\/p>\n

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.<\/p>\n

<\/p>\n

Reasons for dis Continuity of functions<\/strong><\/span><\/h3>\n

(i)\u00a0\"\u00a0 does not exist \u00a0 i.e. \u00a0\"<\/p>\n

(ii) \u00a0f(x) is not defined at x= c<\/p>\n

(iii) \u00a0\"<\/p>\n

Types of Dis Continuity of functions<\/strong><\/span><\/h3>\n

\u00a0Removable type of discontinuities-<\/strong><\/span> In this case\u00a0\"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.<\/p>\n

Removable type of discontinuity can be further classified as:<\/p>\n

(a) Missing Point Discontinuity-<\/strong><\/span> Where f(x) exists finitely but f(a) is not defined.e.g.<\/p>\n

\"f(x)\u00a0here f(x) has a missing point discontinuity at x = 1 , and\u00a0\"f(x)<\/p>\n

has a missing point discontinuity at x = 0<\/p>\n

(b) Isolated Point Discontinuity-<\/strong> Where f(x) exists & f(a) also exists but<\/p>\n

\"\u00a0 \u00a0e.g.\u00a0\"\u00a0here \u00a0\"x\u00a0 & f (4) = 9 has an isolated point<\/p>\n

discontinuity at x = 4. Similarly f(x) = [x] + [ \u2013x] =\u00a0\"\\left[\u00a0 has an isolated point discontinuity at all x\"I.<\/p>\n

Non-Removable type of dis\u00a0Continuity of functions<\/strong><\/span><\/h3>\n

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.<\/p>\n

Non-removable type of discontinuity can be further classified as:<\/p>\n

(a) Finite discontinuity-<\/strong><\/span>\u00a0e.g. f(x) = x – [x] at all integral, \u00a0\"f(x)<\/p>\n

f(x) = at x = 0 and \u00a0 \u00a0\u00a0\"f(x)\u00a0\u00a0at x = 0 \u00a0[note that f(0+<\/sup>) = 0 ; f(0\u2013<\/sup>) = 1]<\/p>\n

(b) Infinite discontinuity-<\/strong><\/span>\u00a0 e.g.\u00a0<\/span>\"f(x)\u00a0\u00a0or \u00a0<\/span>\"g(x)\u00a0\u00a0at x = 4,\u00a0<\/span>\"f(x)<\/p>\n

at x =0 and \u00a0<\/span>\"f(x)\u00a0\u00a0x = 0.<\/span><\/p>\n

(c) Oscillatory discontinuity-<\/strong> <\/span>e.g. \u00a0<\/span>\"f(x)\u00a0\u00a0at 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 \u00a0<\/span>\"\u00a0\u00a0does not exist.<\/span><\/p>\n

4.The Jump Of Discontinuity-<\/strong><\/span> 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<\/em> in this interval.<\/p>\n

5<\/strong>. All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains.<\/p>\n

6.<\/strong> If f & g are two functions that are continuous at x= c then the functions defined by :<\/p>\n

F1<\/sub>(x) = f(x) \u00b1 g(x) \u00a0; \u00a0F2<\/sub>(x) = K f(x) , K any real number \u00a0; F3<\/sub>(x) = f(x).g(x) are also continuous at x= c. Further, if g (c) is not zero, then \u00a0\"{F_4}\\left(\u00a0\u00a0is also continuous at \u00a0 x= c. <\/span><\/p>\n

If you are finding anything difficult to understand, you can take help from IB Maths Tutors<\/strong> <\/a>for free<\/p>\n

Theorems of Continuity\u00a0<\/b><\/span><\/h3>\n

(a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function<\/span>\"\\varphi\u00a0is not necessarily be discontinuous at x = a. e.g. f(x) = x & \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/p>\n

g(x) =\"\\left[<\/span><\/p>\n

(b) If f(x) and g(x) both are discontinuous at x = a then the product function \"\\varphi<\/span><\/p>\n

\u00a0is not necessarily be discontinuous at x = a. e.g \u00a0 \u00a0<\/span>\"{\\rm{f(x)<\/p>\n

(c) Point functions are to be treated as discontinuous. eg.\u00a0\"f(x)\u00a0 is not continuous at x = 1.<\/p>\n

(d) A Continuous function whose domain is closed must have a range also in closed interval.<\/p>\n

(e) If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is<\/p>\n

continuous at x = c. eg. \u00a0\u00a0\"f(x)\u00a0\u00a0& \u00a0\u00a0\"g(x)\u00a0 \u00a0are continuous at x = 0 , hence the<\/p>\n

composite \u00a0\u00a0\"(gof)\u00a0 \u00a0will also be continuous at x = 0<\/p>\n

(f)\u00a0\"f(x)\u00a0this nth degree polynomial is continuous for x\"R<\/p>\n

(g)\u00a0\u00a0y=Sinx, y=Cosx are continuous \u00a0for\u00a0x\"R<\/p>\n

(h) \u00a0<\/strong>y=\u00a0\"{\\log\u00a0is continuous \u00a0for all x>0<\/p>\n

(i)\u00a0<\/strong>\u00a0y=\u00a0\"{a^x}\"\u00a0is continuous \u00a0for all \u00a0 x\"R<\/p>\n

7. Continuity In An Interval<\/strong><\/span><\/h3>\n

(a) A function f is said to be continuous in (a, b) if f is continuous at each & every point \u00a0 \u00a0 \u00ce\"(a, b)<\/p>\n

(b) A function f is said to be continuous in a closed interval [a,b] if:<\/p>\n

(i) f is continuous in the open interval (a, b) &<\/p>\n

(ii) f is right continuous at \u2018a\u2019 i.e. \u00a0\"\u00a0= a finite quantity.<\/p>\n

(iii) f is left continuous at \u2018b\u2019 i.e. \u00a0\u00a0\"\u00a0= a finite quantity. Note \u00a0that a function f which is continuous in possesses the following properties :<\/p>\n

Note \u00a0that a function f which is continuous in [a,b] possesses the following properties:<\/p>\n

(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).<\/p>\n

(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).<\/p>\n

(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).<\/p>\n

Sandwich Theorem or Squeeze Theorem<\/span><\/strong><\/h3>\n

Suppose \u00a0 \u00a0\"f\\left(\u00a0 \u00a0\"\\forall\u00a0 in some interval about c and that f(x) and h(x) approaches the same limit L as approaches c i.e<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \"\u00a0 \u00a0 then \u00a0 \u00a0\"<\/p>\n

This theorem is known as Sandwich Theorem.<\/p>\n

Intermediate Value Theorem<\/b><\/span><\/h3>\n

\u00a0<\/b>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-<\/p>\n

(i) \u00a0 \u00a0\"a<\/p>\n

(ii) \u00a0 f(c)=M<\/p>\n

Download the PDF at practice at least one question on each concept so that you remember them forever<\/span>. In case of any problem take help from Online Maths Tutors<\/strong> for free<\/span><\/p>\n

You can go to the following links to read the posts about limits
\nPost on limits- part one<\/h4>\n

Post on limits-part two<\/h4>\n

\"ib<\/p>\n

Whatsapp aaat +919911262206 or fill the form to get 1 hr free class<\/h5>\n\n
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    <\/p>\n","protected":false},"excerpt":{"rendered":"

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