1.

A real valued function f(x) is given as f(x) = {{:(int_(0)^(x) 2{x}dx",",x + {x} in I),(x^(2) - x + (1)/(2)",",(1)/(2) lt x lt (3)/(2) and x ne "1, where" []),(x^(2) - x + (1)/(6)",","otherwise"):} denotes greatest integer less than or equals to x and {} denotes fractional part function of x. Then,

Answer»

F(X) is continuous and differentiable in `x in [-(1)/(2),(1)/(2)]`
f(x) is continuous and differentiable in `x in [-(1)/(2),(1)/(2)]`
f(x) is continuous and differentiable in `x in [(1)/(2),(3)/(2)]`
f(x) is continuous but not differentiable in `x in (0, 1)`

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