Given `f(x) = 2x, x > 0` ` , 0 , x le 0` then f(x) is ........

Options

A. discontinus and not differentiable at x = 0
B. continuous and differentiable at x = 0
C. discontinous and differentiable at x = 0
D. continuous and not differentiable at x= 0

Explanation

We observe that the function f(x) is not same at x=0 and at all other points of domain.
Here, for `x gt 0" "f(x) = 0`
`" "underset(x to 0^(+))(limf(x)) = underset(x to 0)(lim (0))=0`
`" "underset(x to 0^(-))(lim f(x)) = underset(x to 0)(lim (2x) =0)`
`f(0) = 0`
Hence f(x) is continous at all points .
So, f(x) is continous at all points
Now, let us discuss differentiabiliyt of `f(x)` at x = 0 .
`f(0^(-)) = underset(h to 0)lim(f(0+h)-f(0))/(h)`
` = underset(h to 0)(lim) (2(0+h) -0)/(h)`
` = underset(h to 0)(lim) (2h)/(h) = 2`
`f'(0^(+)) = underset(h to 0)(lim) ((0+h)-0)/(h)`
` = underset(h to 0)(lim) (0-0)/(h) =0`
`rArr f'(0^(-)) ne f'(0^(+)) `
So, f(x) is not differentiable at x = 0 .
`therefore f(x)` is continous and not differentiable at x = 0 .
Hence, the correct answer form the given alternative is (d).