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yuxin91960's Question
Math
Posted about 1 month ago
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)=i=1nj=1naijxixj+k=1nbkxk+cf(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 pij(x)=xixjp_{ij}(x) = x_i x_j defined in (1a) are continuous for all i,j{1,,n}i, j \in \{1, \ldots, n\}
step 3
The linear maps Lk(x)=bkxkL_k(x) = b_k x_k are continuous for each k{1,,n}k \in \{1, \ldots, n\}
step 4
The constant map cc is continuous
step 5
The sum of continuous functions is continuous, thus f(x)f(x) is continuous as it is a finite sum of continuous maps
Answer
The function f(x)f(x) is continuous everywhere in Rn\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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