Exercise 5.2.5

Question

Let $$f_{a}(x)= \begin{cases}x^{a} & 	ext { if } x>0 \ 0 & 	ext { if } x \leq 0\end{cases}$$
  \begin{enumerate}[label=(\alph*)] 
  \item For which values of $a$ is $f$ continuous at zero?
  \item For which values of $a$ is $f$ differentiable at zero? In this case, is the derivative function continuous?
  \item For which values of $a$ is $f$ twice-differentiable?
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item All $a > 0$, if $a = 0$ we get the step function and $a < 0$ gives an asymptote
  \item At zero, if the derivative exists, then the one-sided limit from above
  $$\lim_{x \to 0^+} \frac{x^a}{x} = \lim_{x \to 0^+} x^{a-1}$$
        must exist and be equal to zero (to match the limit from below). Thus $a > 1$ is necessary, and in this case the derivative will be continuous.
  \item The first derivative is $ax^{a-1}$, and its derivative at zero must be zero, so $a > 2$.
   \end{enumerate}

Exercise 5.2.5

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Exercise 5.2.5

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