Need to prove that f(x)=x1/5 is continuous everywhere, where f:R→R:
from definition we need to show that given ϵ>0 ∃δ>0 s.t. |x−x0|<δ⇒|x15−x150|<ϵ for any point x0∈R
I have a proof but it's somewhat unjustified:
consider
|x15−x150|≥|x15|−|x150| from the triangle inequality since |x15|<|x| and |x150|<|x0| then |x15−x150|≥|x15|−|x150|<|x|−|x0|=|x−x0|=δ so we can choose δ=ϵ?
Overall I'm not happy with the proof, in the last inequality I don't think I can just state that delta = epsilon and be done, but I have no idea what else to do. I also am not sure about this step |x|−|x0|=|x−x0|and |x15|<|x| and |x150|<|x0| that step also...
if anyone could help me out.. thank you
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