Question

If f(x) = `{{:(x^2",", x ≥ 0),(x^3",", x < 0):}`, then f(x) is ______.

Options

• continuous on R

• discontinuous on R

• continuous on R except at x = 0

• continuous on R except at x = 0

Answer 1

If f(x) = `{{:(x^2",", x ≥ 0),(x^3",", x < 0):}`, then f(x) is continuous on R.

Explanation:

For x > 0, f(x) = x²

Since f is a polynomial function, it is continuous for all x > 0.

For x < 0, f(x) = x³

Since f is a polynomial function, it is continuous for all x < 0.

`lim_(x -> 0^-) "f"(x) =  lim_(x -> 0) x^3` = 0

`lim_(x -> 0^+) "f"(x) =  lim_(x -> 0^+) x^2` = 0

f(0) = 0

∴ f(x) is continuous at x = 0.

∴ f(x) is continuous on R.