(related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions)
By hypothesis, \(f,g:D\to\mathbb R\) are functions, which are continuous at the number \(a\in\mathbb R\), which means by definition that for any sequence \((x_n)_{n\in\mathbb N}\) of numbers \(x_n\in D\) convergent to \(a\), i.e. with \[\lim_{n\to \infty} x_n=a,\] we have \[\lim_{n\to\infty }f(x_n)=f(a)\text{ and }\lim_{n\to\infty }g(x_n)=g(a).\quad\quad( * )\]
From the proposition about sums of convergent real sequences, it follows that \[\lim_{n\to\infty}(f + g)(x_n)=\lim_{n\to\infty}f(x_n) + \lim_{n\to\infty}g(x_n)\overset{( * )}{=}f(a) + f(b), \] which means that \(f + g:D\to\mathbb R\) is continuous at \(a\).
From the proposition about the product of a real number with a convergent real sequence, it follows that \[\lim_{n\to\infty}(\lambda f)(x_n)=\lambda \lim_{n\to\infty}f(x_n)\overset{( * )}{=}\lambda f(a), \] which means that \(\lambda f:D \to \mathbb R\) is continuous at \(a\).
From the proposition about the product convergent real sequences, it follows that \[\lim_{n\to\infty}(f \cdot g)(x_n)=\lim_{n\to\infty}f(x_n) \cdot \lim_{n\to\infty}g(x_n)\overset{( * )}{=}f(a) \cdot f(b), \] which means that \(f\cdot g:D \to \mathbb R\) is continuous at \(a\).
From the proposition about the quotient of convergent real sequences, it follows that there is a some index \(N\in\mathbb N\) such that \(g(x_n)\neq 0\) for all \(n \ge N\) we have \[\lim_{n\to\infty,n \ge N}\left(\frac fg\right)(x_n)=\frac{\lim_{n\to\infty,n \ge N}f(x_n)}{\lim_{n\to\infty,n \ge N}b(x_n)} \overset{( * )}{=}\frac{f(a)}{f(b)}, \] which means that \(\frac fg:D' \to \mathbb R\) is continuous at \(a\), with \(D':=\{x\in D:g(x)\neq 0\}\).
