Q5: (10 pts) Let K > 0 and f R R satisfying the condition lf(x)-f(y) | Klx-y | for all x, y E R. ...

Question

Question

Answer 1

Answer #1

$CER$

We want to show that given any $\epsilon >0$ there exists a $δ>0$ such that $|x-c|<\delta \Rightarrow |f(x)-f(c)|<\epsilon$

We have for $\epsilon>0$ , choose $\delta=\frac{\epsilon}{2K}$ which is positive

And we have $|x-c|<\delta\Rightarrow |f(x)-f(c)|\leq K|x-c|=K\times \frac{\epsilon}{2K}$

That is, $|x-c|<\delta\Rightarrow |f(x)-f(c)|\leq \frac{\epsilon}{2}<\epsilon$

So that $|x-c|<\delta\Rightarrow |f(x)-f(c)|<\epsilon$

Thus, we have $f is continuous at c$ as we wanted to show

Answer 2

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