Slide 1: Continuity A Power Point Presentation by:
D P Sharma
KV,AFS,Kumbhirgram. Slide 2: Continuity: Sub Titles:
1.Meaning of Continuity an Intuitive idea.
2.Continuity of a real function at a point.
(Graphical Interpretation and definition)
4.Continuity of a real function in a interval(Continuous
5.Algebra of Continuous Function(Statement).
7.Previous year board questions and HOTs.
8.Home work. Slide 3: 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. Slide 4: 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. Slide 5: 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 Slide 6: 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 Slide 7: 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 Slide 8: 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. Slide 9: Types of Discontinuity: 1.Removable Discontinuity: 2.Discontinuity of first kind: 3.Discontinuity of Second Kind: Example Example Example Slide 10: 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 Slide 11: 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) Slide 12: 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 Slide 13: Here, LHL = Similarly RHL=0 & f(2)=0 i.e. LHL=RHL=f(2). f(x) is continuous at x=2. Back Slide 14: And , So function is continuous at x=0 Back Slide 15: Previous year board questions(Model Paper): Sample paper 2009 Model paper G:\cbse2009 Home Work Slide 16: (0,3) Slide 9 Slide 17: Y=f(x) (0,2) (0,1) Slide 9 Slide 18: Y’ Slide 9 Slide 19: Back Slide 20: THANK YOU