1. RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE-411028.
PRESANTATION BY
Prof . DESAI S.S
Mathematics department
Subject – Real analysis
Topic Differntatiable Function

2. Derivable at point :-
Let f be a real valued function defined on interval [a,b]. Then f is said to be Derivable at an interior point c,
if exist.
It is denoted by ,

3. Left hand derivative at x=c :- Right hand derivative at x=c :-
Left hand derivative at x=c is
defined as ,
if this limit exist.
Right hand derivative at x=c :-
Right hand derivative at x=c is
defined as ,
if this limit exist.

4. Differentiablity Implies Continuity :-
Theorem:- If f: I→ ℝ, then function f has the derivative at x=c, where c ∈ I, then f is continuous at x=c.
Corollary:- If f is derivable at all points of an interval, then it is continuous in the interval.
Note:- Converse of above theorem is not true.
e.g ,
This function is continuous at x=0, but not
differentiable at x=0. ℝ

5. Rolle’s Theorem :- Statement:- Suppose that a function f is ,
i) continuous on [a,b],
ii) Differntiable on (a,b) &
iii) f(a)=f(b),
then there exist at least one point c ∈ (a,b), such that

6. Lagranges Mean Value Theorem:-
Statement :-
Suppose that a function f is
continuous on [a,b] &
deriavable on the (a,b) then
there exist at least one point
c ∈ (a,b), such that ,

7. Thank You …