let g(x) be a continous function for all x ,and f(x) = f(a) +(x-a)g(x) for all x ε R , then
a. f(x) is necessarily differentiable at x =a
b. f(x) is not necessarily differentiable at x=a
c. f(x) is not necessarily continous at x=a
d. N . O .T .
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2 Answers
1f'(a^+)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a))}{h}=g(a+h)\\ \\f'(a^-)=\lim_{h\rightarrow 0} \frac{f(a-h)-f(a))}{-h}=g(a-h)\\ \\ as\ g(x)\ is\ cont,\so, \\lim_{h\rightarrow 0}(g(a+h))=\lim_{h\rightarrow 0}(g(a-h))so, f'(a^{+})=f'(a^{-})\\ \\ so\ f(x)\ must\ be\ differentiable\ and\ cont\ at\ x=a
