Linear Function is Continuous

Theorem
Let $$\alpha, \beta \in \mathbb{R}$$ be real numbers.

Let $$f$$ be the real function defined as $$f \left({x}\right) = \alpha x + \beta$$.

Then $$f$$ is continuous at every real number $$c \in \mathbb{R}$$.

Proof

 * First assume $$\alpha \ne 0$$.

Let $$\epsilon > 0$$.

Let $$\delta = \frac {\epsilon}{\left|\alpha\right|}$$.

Then, provided that $$\left|{x - c}\right| < \delta$$:

$$ $$ $$ $$

So, we have found a $$\delta$$ for a given $$\epsilon$$ so as to make $$\left|{f \left({x}\right) - f \left({c}\right)}\right| < \epsilon$$ provided $$\left|{x - c}\right| < \delta$$.

So $$\lim_{x \to c} f \left({x}\right) = f \left({c}\right)$$ and so $$f$$ is continuous at $$c$$, whatever $$c$$ happens to be.


 * Now suppose $$\alpha = 0$$.

Then $$\forall x \in \mathbb{R}: f \left({x}\right) - f \left({c}\right) = 0$$.

So whatever $$\epsilon > 0$$ we care to choose, $$\left|{f \left({x}\right) - f \left({c}\right)}\right| < \epsilon$$, and whatever $$\delta$$ may happen to be is irrelevant.

Continuity follows for all $$c \in \mathbb{R}$$, as above.