continuity

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Continuity A Power Point Presentation by: D P Sharma PGT(Maths) KV,AFS,Kumbhirgram.

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Continuity: Sub Titles: 1.Meaning of Continuity an Intuitive idea. 2.Continuity of a real function at a point. (Graphical Interpretation and definition) 3.Discontinuity(Types) 4.Continuity of a real function in a interval(Continuous Function). 5.Algebra of Continuous Function(Statement). 6.Illustrative Examples. 7.Previous year board questions and HOTs. 8.Home work.

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1.Meaning of Continuity an Intuitive idea. 1.The term ‘Continuity’ in math has not more different meaning than its meaning in our day –to-day life. *When someone says, “Ram is running.” Certainly its indicate that there is continuation in the work(activity).Let the running track is elliptical. Then the curve traced by Ram will be as below: The path is continuous. There are a lot of example in our surrounding: 1.Flowing of water in the river. 2.Running of trains on their track etc.

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2.Continuity of a real function at a point. (Graphical Interpretation and definition) n a b c X When we say a function f(x)is continuous at a point x=a it means that at point (a,f(a)) the graph of the function has no holes or gaps. Let f(x) is a function whose graph is as shown in the above figure. Here we observe that there are three points x=a, x=b and x=c where the function is not continuous.

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At point x=a :It is seen that there is a hole in the graph of f(x) corresponding to point x=a. So,the curve y=f(x) is not continuous at x=a. Here at x=a,f(x) is not defined. Graph Thus the continuity of the function at x=a Can be destroyed if the limit of f(x) at x=a Exists but f(x) is not defined at x=a. Back

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At point x=b :From the graph of the function f(x) it is evident that Graph Thus continuity of a function f(x) at x=b can also be destroyed if Back

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At point x=c: at point x=c,the curve y=f(x) is not Continuous,because Graph So,the continuity of a function f(x) can also be destroyed if Back

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From these discussions the continuity and discontinuity of a function at point can be defined in this way: Definition: A function f(x) is said to be continuous at a point x=a , iff If f(x) is not continuous at a point x=a, then it is said to be discontinuous at x=a.

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Types of Discontinuity: 1.Removable Discontinuity: 2.Discontinuity of first kind: 3.Discontinuity of Second Kind: Example Example Example

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Continuity of a real function in an interval. A real function f is said to be continuous if it is continuous at every point in the domain of f. Suppose f is a function defined on a closed interval ,then for f to be continuous ,it need to be continuous at every point in including the end points Continuity of f at ‘a’ means Continuity of f at ‘b’ means

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Algebra of Continuous Function(Statement). *Suppose f and g be two real functions continuous at a real point ‘c’. Then. 1.f+g is continuous at x=c. 2.f-g is continuous at x=c. 3.f . g is continuous at x=c. 4.f/g is continuous at x=c.(Provided g(c) 0)

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Illustrative Examples. f(x)= if x = 0 1.Check the continuity of the function f at x=0 2. Show that the function is continuous at x=2. 3.Examine the continuity of f where f is defined by f(x)= sin x –cos x if x 0 -1 if x = 0 4. If the function : 3ax+b if x > 1 f(x) = 11 if x = 1 5ax-2b if x < 1   is continuous at x = 1, find the values of a and b. Solution Solution

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Here, LHL = Similarly RHL=0 & f(2)=0 i.e. LHL=RHL=f(2). f(x) is continuous at x=2. Back

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And , So function is continuous at x=0 Back

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Previous year board questions(Model Paper): Sample paper 2009 Model paper G:\cbse2009 Home Work

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(0,3) Slide 9

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Y=f(x) (0,2) (0,1) Slide 9

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Y’ Slide 9

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Back

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THANK YOU

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