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QUESTION: Hi Randy,

Here is my last question. I am confused over questions like "Describe where the following functions are continuous?". I am confused that for those questions like y=(1)/(1+2^1/3) we only look at the Domain (-inf,-1/8) union(-1/8, inf) and than state that the function is continuous at the x-axis because the composite function is composed of two continuous function. My question is do we not need to look at if the function is also continuous on the y-axis and when asked is the function continuous do we only need to look at the x-axis/domain.

Many Many Thanks,
Sam

ANSWER: Your formula y = 1/(1+2^1/3) looks a little funny; where's the x?

The union of the 2 domains in x should include x = -1/8. I'm not sure if your notation does this.

---------- FOLLOW-UP ----------

QUESTION: Sorry about that.

The equation should be y = 1/(x+2^1/3) and I would like to know how do I prove that this function is continuous algebraically. Is it just by looking at the domain of the two composites that this function can break down to?

Many Thanks,
Sam

Answer
Given that (-1/8)^1/3 = -1/2, the denominator of your expression should probably be something like x^1/3+1/2 so that it is singular at x = -1/8.  Not sure. But anyway, the definition of the domain includes the entire real line except x=-1/8; the parenthese (...) usually denote an open interval, i.e., does not include the end point, the square brackets [...] denote a closed interval. So the function is continuous on each side of this value (no singularity). There are no other singularities in these 2 disjoint (good term; look it up) domains so it can be designated continuous.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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