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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)?

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    I supposed you mean $D$ as a domain. So, its not correct to write $D\in\mathbb{R}$. You should write it as $D\subseteq\mathbb{R}$. – 2017-01-16

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

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

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    Can you elaborate? I didn't quite understand. – 2017-01-16
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    Does it mean that we are doing the exact same thing (except that we substitute $3$ with $x_0$). So the result is the same? – 2017-01-16