How can I ensure continuity for a piecewise function with a radical term?

[Petrus]

Hello MHB,
If I want to decide constant a and b so its continuous over the whole R for this piecewise function

basicly what I got problem with is that \(\displaystyle x^{1/3}\) is not continuous for negative value so it will never be continuous for any value on constant a,b. I am missing something? or do they mean \(\displaystyle \frac{1}{8}\) and not \(\displaystyle -\frac{1}{8}\)

Regards,
\(\displaystyle |\pi\rangle\)

 

[Amer]

Hello MHB,
If I want to decide constant a and b so its continuous over the whole R for this piecewise function

basicly what I got problem with is that \(\displaystyle x^{1/3}\) is not continuous for negative value so it will never be continuous for any value on constant a,b. I am missing something? or do they mean \(\displaystyle \frac{1}{8}\) and not \(\displaystyle -\frac{1}{8}\)

Regards,
\(\displaystyle |\pi\rangle\)

In fact
$$\sqrt[3]{x}$$ is defined for all real numbers but the problem is in
$$\sqrt[n]{x}$$ with n even number
for a function to be continuous at a point c
$$\lim_{x \rightarrow c^- } f(x) = \lim_{x\rightarrow c^+ } f(x) = f(c)$$

 

[Petrus]

In fact
$$\sqrt[3]{x}$$ is defined for all real numbers but the problem is in
$$\sqrt[n]{x}$$ with n even number
for a function to be continuous at a point c
$$\lim_{x \rightarrow c^- } f(x) = \lim_{x\rightarrow c^+ } f(x) = f(c)$$

Thanks for the fast respond you are totally correct! I confused myself! Have a nice day!

Regards,
\(\displaystyle |\pi\rangle\)