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im2fastfouru
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There is a proof in my book that asks us to prove that the product of two continuous functions is continuous. If anyone could help please reply back, thanks!
lurflurf said:hint
write
f(x+h)=f(x)+[f(x+h)-f(x)]
g(x+h)=g(x)+[g(x+h)-g(x)]
note
|f(x+h)-f(x)|<eps1
|g(x+h)-g(x)|<eps2
|f(x+h)-f(x)|,|g(x+h)-g(x)|<eps=max(eps1,eps2)
also recall
|a+b+c|<|a|+|b|+|c|
To solve the problem and because it is fun. I agree, I provided a different (though very slightly) view.sutupidmath said:Why on Earth would he do so? Halls hints are quite straightforward.
theorem f is continuous if and only ifHallsofIvy said:lurflurf may has misread "continuous" as "differentiable".
A useful frameworkmathwonk said:the point (of continuity) is simply that if two numbers are respectively near two other numbers, then the products are also near each other.
to see this, let the numbers be a+h and b+k and compare the product of ab to that of (a+h)(b+k), when h and k are small.
The product of two continuous functions is a new function that is obtained by multiplying the outputs of two different continuous functions for every input value.
In order for the product of two continuous functions to be continuous, both individual functions must be continuous at the same point. This means that the limit of each function as the input approaches a certain value must exist and be equal.
Yes, it is possible for the product of two continuous functions to be discontinuous. This can happen if one or both of the individual functions have a discontinuity at a certain point, or if the limit of the product function at that point does not exist.
If one of the functions is not continuous, then the product function will also not be continuous. This is because a single point of discontinuity in one of the functions will affect the overall continuity of the product.
Yes, there are a few properties that hold for products of continuous functions. For example, the product of two even or two odd functions will also be even, while the product of an even and an odd function will be odd. Additionally, the product of two periodic functions with different periods will not be periodic.