Step 3, why is Lk(x) continuous

`Write a proof by component wise continuity to show f(x) is continuous`

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Answer from Sia

Posted about 1 month ago

Solution by Steps

step 1

To show that the polynomial $f(x) = \sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij} x_i x_j + \sum_{k=1}^{n} b_k x_k + c$ is continuous, we can express it as a composition of continuous maps

step 2

The maps $p_{ij}(x) = x_i x_j$ defined in (1a) are continuous for all $i, j \in \{1, \ldots, n\}$

step 3

The linear maps $L_k(x) = b_k x_k$ are continuous for each $k \in \{1, \ldots, n\}$

step 4

The constant map $c$ is continuous

step 5

The sum of continuous functions is continuous, thus $f(x)$ is continuous as it is a finite sum of continuous maps

Answer

The function $f(x)$ is continuous everywhere in $\mathbb{R}^n$.

Key Concept

Continuity of Polynomials

Explanation

Polynomials are continuous functions because they can be expressed as compositions and sums of continuous functions, such as linear and constant maps.

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