If `f(x)=x` for `x le 0=0` for `x t 0`, the 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
LHS at x=0 `underset(x to 0^(-))(lim)f(x)=underset(xto0)(lim)x`
`underset(hto0^(+))(lim)f(x)=underset(xto0^(+))(lim)0=0`
Also, f(0)=0
`thereforeLHL=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 `gt0`
`thereforef(x)` is not differentiable at x=0