Prove that if f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n, then f is continuous at a. Hint: Apply a theorem about linear functions, that we covered in class (Bartle p. 149).     For the following exercises, use the following definition: A function f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n if there exists a linear transformation g: ℝ^n → ℝ^m such that lim [f(a+h)-f(a)-g(h)]/h =0 h→0

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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Prove that if f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n, then f is continuous at a. Hint: Apply a theorem about linear functions, that we covered in class (Bartle p. 149).

 

 

For the following exercises, use the following definition:
A function f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n if there exists a linear transformation
g: ℝ^n → ℝ^m such that
lim [f(a+h)-f(a)-g(h)]/h =0
h→0

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