Advanced Math/Continuity (Beginner Calculus)
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,
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.
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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?
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.