Define `f` on `R` into itself by `f(x)={(x"sin"1/x, "when"x!=0),(0,"when"x=0):}` then

Options

A. `f` is continuous at 0 but not differentiable at 0
B. `f` is both continuous and differentiable at 0
C. `f` is differentiable but no continuous at 0
D. None of the above

Answer

A

Explanation

LHL ltbRgt `=lim_(xto0^(-))f(x)=lim_(hto0)-h sin (1/(-h))=0`
`RHL=lim_(xto0^(+))=lim_(hto0)h sin (1/h)=0`
`:.LHL=RHL=f(0)`, it is continuous.
LHD
`=lim_(xto0^(-))f(x)=lim_(hto0)[(f(0-h)-f(0))/(-h)]`
`=lim_(hto0)(-h"sin"1/h-0)/(-h)=` does not exist
`impliesf(x)` is not differentiable at `x=0`
`:.f(x)` is continuous at `x=0` but not differentiable at `x=0`