The function f(x) = x |x| is ______.

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KollegeApply · Accepted Answer

The function f(x) = x |x| is continuous and differentiable at x = 0. Explanation: Given, the function is f(x) = x |x| for x ∈ R. The function can be written as, f(x) = `{{:( x^2; x > 0),(-x^2; x $\leq$ 0):} Now, Rf(0) = lim_(x rightarrow 0^+) (x^2) = 0 and Lf(0) = lim_(x rightarrow 0^-) (-x^2)` = 0 So, Lf(0) = Rf(0) = f(0) So, the function is continuous at 0. Now, Rf'(0) = `lim_(x rightarrow 0^+) (f(x) - f(0))/(x - 0) = lim_(x rightarrow 0^+) (x^2 - 0)/x = 0 and Lf'(0) = lim_(x rightarrow 0^-) (f(x) - f(0))/(x - 0) = lim_(x rightarrow 0^-) (-x^2 - 0)/x` = 0 So, Lf'(0) = Rf'(0) So, function is differentiable at 0.

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