If f(x)=cubed root of x, how do you determine if f is continuous and/or differentiable at 0 and if it's f(0)=0?

Question

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Answer 1

Zero times itself is 0. Zero times itself again is also 0. Therefore 0*0*0 = 0 and so its cube root is also 0.

further, f(x) = third root of x. This function is continuous everywhere which you can verify by plotting it as every number, including zero, has a third root. In the case of zero that number is itself and with nonzero numbers it will be some decimal or integer depending on the inputted number. This includes negative numbers since any three negative numbers multiplied together will return a negative number, thus all negative numbers have a third root.

The derivative of the third root of x is 1/3*(x^-2/3) which can be calculated using the chain rule. Evaluating this at 0 returns

1/3(0^-2/3) note that raising a number to the negative 2/3rds power is equivalent to taking the third root of that number squared in the denominator. 0 squared is 0 and the third root of 0 is 0 multiply this by three which gives zero in the denominator or 1/0 overall thus it is not differentiable at 0.