# Proof

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 $\lim_{n\to\infty }f(x_n)=f(a)\text{ and }\lim_{n\to\infty }g(x_n)=g(a).\quad\quad( * )$

### $$(1)$$ We show that $$f + g:D\to\mathbb R$$ is continuous at $$a$$.

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 $$a$$.

### $$(2)$$ We show that $$\lambda f:D \to \mathbb R$$ 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 $$a$$.

### $$(3)$$ We show that $$f\cdot g:D \to \mathbb R$$ 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 $$a$$.

### $$(4)$$ We show that for $$g(a)\neq 0$$ the function $$\frac fg:D' \to \mathbb R$$ is continuous at $$a$$, where $$D':=\{x\in D:g(x)\neq 0\}$$.

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 $$D':=\{x\in D:g(x)\neq 0\}$$.

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983