Question

If f(x) = `|x|/x`, for x ≠ 0

= 1, for x = 0, then the function is ______.

Options

• differentiable but not continuous at x = 0

• continuous and differentiable at x = 0

• neither continuous nor differentiable at x = 0

• continuous but not differentiable at x = 0

Answer 1

If f(x) = `|x|/x`, for x ≠ 0

= 1, for x = 0, then the function is neither continuous nor differentiable at x = 0.

Explanation:

We have,

f(x) = `{(|x|/x "," x ne 0),(1  ","  x = 0):}`

For continuous at x = 0

`lim_(x->0^-) "f"(x) = lim_(x->0^+) "f"(x) = "f"(0)`

`lim_(x->0^-) "f"(x) = lim_(x->0 - h) "f"(0 - "h")`

`lim_(h->0) (|0 - "h"|)/(0 - "h") = "h"/(- "h")` = - 1

`lim_(x ->0^-)` f(x) ≠ f(0) = 1

∴ f(x) is discontinuous at x = 0

∴ function is neither continuous nor differentiable at, x = 0