How do we prove that f ( 1 n + 1 ) = x n and f ( 0 ) = x is continuous.

Question

How do we prove that   f      (          1              n        +        1              )    =      x          n       and   f  (  0  )  =  x is continuous.

Sage Mcdowell · Accepted Answer

With that metric, every point of С other than 0 is an isolated point, and therefore every function from С into a metric space (X,d) is continuous at every point other than 0.
You proved correctly that, if f is continuous, then

lim_{n \to \infty} x_{n} = x

. Now, suppose that

lim_{n \to \infty} x_{n} = x

. Take

ε > 0

. Then there is some

N \in N

such that

n ⩾ N ⟹ | x - x_{n} | < ε

. Take

ε > 0

. Then there is some

N \in N

such that

n ⩾ N ⟹ | x - x_{n} | < ε

. And there is a

δ > 0

such that

\frac{1}{n + 1} < δ ⟹ n ⩾ N

such that

\frac{1}{n + 1} < δ ⟹ n ⩾ N

. And there is a

δ > 0

such that

\frac{1}{n + 1} < δ ⟹ n ⩾ N

. Therefore

| \frac{1}{n + 1} - 0 | < δ ⟹ | x - x_{n} | < ε ⟺ | x - f (\frac{1}{n + 1}) | < ε,

and so f is continuous at 0.

How do we prove that f ( 1 n + 1 </mrow>

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Answer & Explanation

New Questions in Calculus 2