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

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 ,

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.

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

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

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 ,

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