Linear Function is Continuous/Proof 2

Theorem

Let $\alpha, \beta \in \R$ be real numbers.

Let $f : \R \to \R$ be a linear real function:

$\map f x = \alpha x + \beta$

for all $x \in \R$.

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

Proof

Let $c \in \R$.

Let $\sequence {x_n}$ be a real sequence converging to $c$.

Then:

\(\ds \lim_{n \mathop \to \infty} \map f {x_n}\)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \paren {\alpha x_n + \beta}\)

\(\ds \)

\(=\)

\(\ds \alpha c + \beta\)

Combined Sum Rule for Real Sequences

\(\ds \)

\(=\)

\(\ds \map f c\)

We therefore have:

for all real sequences $\sequence {x_n}$ converging to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.

So by Sequential Continuity is Equivalent to Continuity in the Reals $f$ is continuous at $c$.

As $c \in \R$ was arbitrary, $f$ is continuous on $\R$.

$\blacksquare$