Continuity of a function for all points

Question

Let $D\in \mathbb{R}$ and $f(x)=5x+7$

a) Show that $f$ is continuous at $x_{0}=3$.

We are asked to use the epsilon delta definition of continuity. So I have the following

$$\lvert 5x+7-22 \lvert =\lvert 5x-15 \lvert = 5 \lvert x-3 \lvert\ < \varepsilon$$

hence $\delta= \displaystyle \frac{\varepsilon}{5}$.

b) Show that $f$ is continuous for all $x_{0} \in \mathbb{R}$

Is a) correct and how do I show b)?

Answer 1

Your proof of $a)$ is O.K.

$b)$ Observe that $|f(x)-f(x_0)|=5|x-x_0|$ and proceed as in a).