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Polter19
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Homework Statement
Hey guys, I've been given the following [itex] \epsilon - \delta [/itex] proof question. The trouble I'm having is I'm not 100% sure what it is actually asking for, and how to go about it. The more common proofs involving two variable functions and a given limit are easy enough to do, but I'm stuck on this one.
Homework Equations
Suppose [itex] f: \left( \mathbb R ^2 , \| . \|_2 \right) \rightarrow \left( \mathbb R , | .| \right) [/itex] is continuous at [itex] \left(a,b \right) \in \mathbb R^2 [/itex]. Prove using the [itex] \epsilon - \delta [/itex] definition only, that if we define the function
[itex] f_b : \left( \mathbb R,|.| \right) \rightarrow \left( \mathbb R,|.| \right) [/itex] s.t. [itex] f_b \left(x \right) = f \left( x,b \right), [/itex]
then [itex] f_b [/itex] is continuous at [itex] x=a [/itex].
The Attempt at a Solution
Well, considering I am not actually sure what the question is acting for I can't go very far. I was assuming you would set up the usual proof as,
Let [itex] \epsilon > 0 [/itex]
if [itex] |(x,b) - (a,b)| < \delta [/itex]
then [itex] |f(x,b) - (a,b)| < \epsilon [/itex]
Possibly then, [itex] x - a < \delta [/itex]?
Any help would be appreciated, cheers.
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