Combination Theorem for Limits of Functions/Real

Theorem
Let $$X$$ be one of the standard number fields $$\Q, \R, \C$$.

Let $$f$$ and $$g$$ be functions defined on an open subset $$S \subseteq X$$, except possibly at the point $$c \in S$$.

Let $$f$$ and $$g$$ tend to the following limits:


 * $$\lim_{x \to c} f \left({x}\right) = l, \lim_{x \to c} g \left({x}\right) = m$$

Let $$\lambda, \mu \in X$$ be any point in $$X$$.

Then the following results hold:

Sum Rule

 * $$\lim_{x \to c} \left({f \left({x}\right) + g \left({x}\right)}\right) = l + m$$;

Multiple Rule

 * $$\lim_{x \to c} \left({\lambda f \left({x}\right)}\right) = \lambda l$$;

Combined Sum Rule

 * $$\lim_{x \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$$;

Product Rule

 * $$\lim_{x \to c} \left({f \left({x}\right) g \left({x}\right)}\right) = l m$$;

Quotient Rule

 * $$\lim_{x \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \frac l m$$, provided that $$m \ne 0$$.

Continuity
If $$f$$ and $$g$$ are continuous on an open subset $$S \subseteq X$$, and $$\lambda, \mu \in X$$, then:


 * $$f + g $$ is continuous on $$S$$;
 * $$\lambda f$$ is continuous on $$S$$;
 * $$\lambda f + \mu g $$ is continuous on $$S$$;
 * $$f g$$ is continuous on $$S$$;
 * $$\frac {f} {g}$$ is continuous on $$S$$.

Proof
These results follow directly from the Combination Theorem for Sequences and Limit of Function by Convergent Sequences, as follows:

Let $$\left \langle {x_n} \right \rangle$$ be any sequence of points of $$S$$ such that $$\forall n \in \N^*: x_n \ne c$$ and $$\lim_{n \to \infty} x_n = c$$.

By Limit of Function by Convergent Sequences, $$\lim_{n \to \infty} f \left({x_n}\right) = l$$ and $$\lim_{n \to \infty} g \left({x_n}\right) = m$$.

By the Combination Theorem for Sequences:
 * $$\lim_{n \to \infty} \left({f \left({x_n}\right) + g \left({x_n}\right)}\right) = l + m$$;
 * $$\lim_{n \to \infty} \left({\lambda f \left({x_n}\right)}\right) = \lambda l$$;
 * $$\lim_{n \to \infty} \left({\lambda f \left({x_n}\right) + \mu g \left({x_n}\right)}\right) = \lambda l + \mu m$$;
 * $$\lim_{n \to \infty} \left({f \left({x_n}\right) g \left({x_n}\right)}\right) = l m$$;
 * $$\lim_{n \to \infty} \frac {f \left({x_n}\right)} {g \left({x_n}\right)} = \frac l m$$, provided that $$m \ne 0$$.

Applying Limit of Function by Convergent Sequences again, we get:


 * $$\lim_{x \to c} \left({f \left({x}\right) + g \left({x}\right)}\right) = l + m$$;
 * $$\lim_{x \to c} \left({\lambda f \left({x}\right)}\right) = \lambda l$$;
 * $$\lim_{x \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$$;
 * $$\lim_{x \to c} \left({f \left({x}\right) g \left({x}\right)}\right) = l m$$;
 * $$\lim_{x \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \frac l m$$, provided that $$m \ne 0$$.

Proof of Continuity
We have that:
 * The complex plane is a metric space;
 * The real number line is a metric space;
 * The rational numbers form a metric space.

Hence it is appropriate to use the defintion of continuity on a metric space.

It follows trivially from the definitions of continuity at a point and continuity on a metric space that, if $$f$$ and $$g$$ are continuous on $$S$$ that the assertions:


 * $$f + g $$ is continuous on $$S$$;
 * $$\lambda f$$ is continuous on $$S$$;
 * $$\lambda f + \mu g$$ is continuous on $$S$$;
 * $$f g$$ is continuous on $$S$$;
 * $$\frac {f} {g}$$ (where $$g \ne 0$$) is continuous on $$S$$

are true.