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
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Prove that if f: ℝ^n → ℝ^m is
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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