Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and $a, b \in \mathbb{R}$. Prove that $g(x) = f(a x + b)$ is continuous.
$f$ is continuous at $a c + b$, so for every $\epsilon > 0$ there exists a $\delta' > 0$ such that $|x - (c + b)| < \delta'$ implies $|f(x) - f(a c + b)| < \epsilon$.
Now, to prove that $g$ is continuous let $\epsilon > 0$, we use $\delta = \frac{\delta'}{|a|}$, so:
$|x - c| < \delta = \frac{\delta'}{|a|}$
$|a| |x - c| < \delta'$
$|(a x + b) - (a c + b)| < \delta'$
So:
$|g(x) - g(c)| = |f(a x + b) - f(a c + b)| < \epsilon$
Is this correct?