Chapter 13

Some Fundamentals∗

This section contains the proofs of the theorems which were stated without proof alongwith some other significant topics which will be useful later. These topics are of funda-mental significance but are difficult.

13.1 Combinations Of Continuous FunctionsTheorem 13.1.1 The following assertions are valid.

1. The function af + bg is continuous at x when f , g are continuous at x ∈ D(f)∩D(g) and a,b ∈ R.

2. If and f and g are each real valued functions continuous at x, then f g is continuousat x. If, in addition to this, g(x) ̸= 0, then f/g is continuous at x.

3. If f is continuous at x, f (x) ∈D(g)⊆Rp, and g is continuous at f (x) , then g ◦fis continuous at x.

4. If f = ( f1, · · · , fq) : D(f)→ Rq, then f is continuous if and only if each fk is acontinuous real valued function.

5. The function f : Rp→ R, given by f (x) = |x| is continuous.

Proof: Begin with (1). Let ε > 0 be given. By assumption, there exist δ 1 > 0 suchthat whenever |x−y|< δ 1, it follows |f (x)−f (y)|< ε

2(|a|+|b|+1) and there exists δ 2 > 0such that whenever |x−y| < δ 2, it follows that |g (x)−g (y)| < ε

2(|a|+|b|+1) . Then let0 < δ ≤ min(δ 1,δ 2). If |x−y| < δ , then everything happens at once. Therefore, usingthe triangle inequality

|af (x)+bf (x)− (ag (y)+bg (y))|

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