Question

If f(x) = `{{:(x, "for"  x ≤ 0),(0,
"for"  x > 0):}`, then f(x) at x = 0 is ______.

Options

• continuous but not differentiable

• not continuous but differentiable

• continuous and differentiable

• not continuous and not differentiable

Answer 1

If f(x) = `{{:(x, "for"  x ≤ 0),(0,
"for"  x > 0):}`, then f(x) at x = 0 is continuous but not differentiable.

Explanation:

We have, f(x) = `{{:(x",", "for"  x ≤ 0),(0",",
"for"  x > 0):}`

LHL at x = 0 = `lim_(x rightarrow 0^-)` f(x) = `lim_(x rightarrow 0)` x

and RHL at x = 0 = `lim_(x rightarrow 0^+)` f(x) = `lim_(x rightarrow 0)` 0 = 0

Now, f(0) = 0

So, LHL = RHL = f(0)

Hence, f(x) is continuous at x = 0

Here, f'(x) = `{{:(1  "for"  x ≤ 0),(0  "for"  x > 0):}`

Thus, f(x) is not differentiable at x = 0