# prove that function f() is continous when $|(f(x)-f(y))|\leq 15|x-y|$

Let f be a function from real numbers to real numbers. The following holds for the function:

$|(f(x)-f(y))|\leq 15|x-y|$

How do I prove that the function is continuous?

(Utilizing the limit concept of continuity).

$f$ is continuous in $x_0$ if $$\forall \varepsilon>0 \ \exists \delta : |x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|\leq \varepsilon$$

So, fix $\varepsilon>0$. We have $|f(x)-f(x_0)|\leq15|x-x_0|$, so it's enough to pick $\delta=\frac{\varepsilon}{15}$:

$$|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|\leq15|x-x_0|\leq 15 \delta=15\frac{\varepsilon}{15}=\varepsilon$$

For some intuition, fix a point $f(x_0)$. This condition says that $|f(x)-f(x_0)|\leq15|x-x_0|$. Eliminating absolute values, we get $$f(x_0)-15|x-x_0|<f(x)<f(x_0)+15|x-x_0|$$ so that means $f(x)$ should be between the cone formed by the lines $f(x_0)+15(x-x_0)$ and $f(x_0)-15(x-x_0)$, as can be seen in the image (for $x_0=0$): • Could you give some intutition on how this condition implies continuity? (On a graph, perhaps) – The Cryptic Cat Feb 21 '17 at 0:26
• @flytothesurface Thank you. Should there not be a "for all of x" quantifier in the definition of continuous? – Dole Feb 21 '17 at 0:54
• @Dole you mean in like $$\forall \varepsilon >0 \ \exists \delta>0: \forall x: |x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\varepsilon$$ If it's yes, it's implied in the "$\Rightarrow$", it says that "whenever $|x-x_0|<\delta$, then... But I'm aware some authors write it. – A. Salguero-Alarcón Feb 21 '17 at 17:16
• @TheCrypticCat I edited my answer according to your comment. – A. Salguero-Alarcón Feb 21 '17 at 17:31
• @flytothesurface Thanks! – The Cryptic Cat Feb 22 '17 at 15:10