Show with $\varepsilon$-$\delta$ that the function $f(x) = \frac{2x+3}{5}$ is continuous

Question

Show with $\varepsilon$-$\delta$ that the function $$f(x) = \frac{2x+3}{5}$$ is continuous.

I found this task on the internet but it was without solution, so I'm asking here if I did it correctly.

Let $\varepsilon>0$ and let $\delta = \frac{\varepsilon}{2}$. If $|x-x_{0}|< \delta$, then:

$$\left |\frac{2x+3}{5} - \left(\frac{2x_{0}+3}{5}\right)\right | = \left|\frac{2x+3-(2x_{0}+3)}{5}\right|=\left|\frac{2x+3-2x_{0}-3}{5}\right|$$

$$=\left|\frac{2x-2x_{0}}{5}\right|<|2x-2x_{0}|=\left|2(x-x_{0})\right|<|2\delta|=\varepsilon$$

Therefor, the function is continuous.

---

Is it alright?

I'm not sure because I have just removed $5$ from the denominator and claimed it's greater without (it really is..). But am I actually allowed to do that?

Answer 1

Your proof is complete. Are you doubting that whenever $A < B$ and $B < C$, then also $A < C$ ? Don't :)