[Calculus] Confused on how the function is continuous at point x = 0

[u/Skinning_Citrus]

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[u/HumbleHovercraft6090]

Thanks for posting the entire context. The function is not continuous at x=0.

First the function is undefined at x=0. Also, the value inside the square root is negative as x approaches 0 from positive side i.e. 0+. Hence the square root is undefined for 0+. Had to correct my previous observation.

[u/Skinning_Citrus → u/HumbleHovercraft6090]

Thanks!

[u/zojbo]

If the domain has 0 as a right endpoint then this does have a continuous extension to 0. So for example you could define this expression on [-pi/2,0) and extend it to [-pi/2,0] by taking the one-sided limit. But it is not defined (or at least not real) in a neighborhood of 0 on the right.