If `f(x) = x` for `x le 0 = 0 ` for `x gt 0`, then `f(x)` at `x = 0` is

Options

A. continuous but not differentiable
B. not continuous but differentiable
C. continuous and differentiable
D. not continuous and not differentiable

Answer

A

Explanation

Given, `f(x)={{:(x,"for"xle0),(0,"for"xgt0):}`
For continuity at x = 0
`underset(hrarr0^(-))("lim") f(x) = underset(xrarr0)("lim") x`
`underset(hrarr0^(-))("lim")(0-h) = 0` and RHL at x = 0
`underset(xrarr0^(+))"lim" f(x) = underset(xrarr0^(+))("lim") 0 = 0`
Also, `f(0) = 0`
`:.` LHL = RHL = `f(0)`
Hence, `f(x)` is continuous at `x = 0`
For differentiability at `x = 0`
`f'(x) = 1` for `x le 0, 0` for `x gt 0`
`:. f(x)` is not differentiable at `x = 0`