CONTINUITY & DIFFERENTIABILITY

1. CONTINUITY & DIFFERENTIABILITY Presented by Muhammad Sarmad Hussain Noreen Nasar

2. Overview • Continuity • Differentiability

3. Continuous at a Point • A function f is continuous at point a in its domain, if exists &

4. Domain • If the point a is not in the domain of f, we do not talk about whether or not f is continuous at a.

5. Continuous on Subset of Domain • The function f is continuous on a subset S of its domain, if it is continuous at every point of the subset.

6. EXAMPLES • CLOSED FORM FUNCTIONS • DOMAIN SPECIFIC FUNCTIONS • NON-CLOSED FORM FUNCTIONS

7. Closed Form Functions All functions are continuous on their (whole) domain. A closed-form function is any function that can be obtained by combining • constants, • powers of x, • exponential functions, • logarithms, and trigonometric functions

8. Closed Form Functions Examples of closed form functions are:

9. Domain Specified Functions The function f(x) = 1/x is continuous at every point of its domain. Note that 0 is not a point of the domain of f, so we don't discuss what it might mean to be continous or discontinuous there.

10. Non – Closed Form Functions The function f(x)   = -1 if x ≤ 2 if x > 2 is not a closed-form function Because we need two algebraic formulas to specify it. Moreover, it is not continuous at x = 2, since limx 2f(x) does not exist

11. Discontinuous

12. Discontinuous Discontinuous Discontinuos

13. Graphical Representation

14. Graphical Representation Continuous

15. Graphical Representation Continuous Undefined

16. Graphical Representation Continuous Discontinuous Undefined

17. Graphical Representation

18. Graphical Representation Discontinuos Continuous Undefined

19. Graphical Representation

20. Graphical Representation Continuous but not Differentiable

21. Graphical Representation Differentiable Continuous but not Differentiable

22. Differentiable • f is differentiable at point “a” if f’(a) exists • f is differentiable on the subset of its domain if it is differentiable at each point of the subset.

23. Undifferentiable • f is not differentiable when • The limit does not exist, i.e. does not exist. This situation, when represented in graphical form leads to a cusp in the graph

24. Undifferentiable

25. Undifferentiable • Limit goes to infinity, e.g. in the following case

26. Isolated Non – Differentiable Points • Consider the following examples

27. Isolated Non – Differentiable Points

28. Difference of Opinion Domain of f(x) is THE bone of contention • f(x) is differentiable throughout its domain • f(x) is not differentiable throughout its domain

29. The Question • Are all continuous functions differentiable ?

30. Another Question • If f is not continuous at point a, then is it differentiable at that point ?

31. One More • Are all differentiable functions continuous ? YES • Mathematically provable and easy to understand.

32. Proof of Continuity • Suppose f(x) is differentiable at x=a

33. Contd….. So we can rewrite the equation as

34. Continuity exists &

35. Confusions Questions

36. THANKS