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3
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Proving integral of a continuous function is continuous
Let $U \subset \mathbb{R}^N$ be an open set, let $f : U \times [a, b] \to \mathbb{R}$ be a continuous function. Consider the function $$g(x):= \int_a^b f(x,y) \,dy$$
with $x \in U$.
i) Prove ...
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1
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0
answers
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Looking for a sigmoid-like function with different properties
I am looking for a function that is 0 at 0, 1 at 1, increases to a predetermined $x_1$ relatively quickly, acquires a derivative close to 0 (but doesn't actually plateau) then starts increasing at $...
- 3,439
0
votes
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answer
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Is it possible for a function to be continuously derivable over its entire open domain except for a removable discontinuity?
For example, is there a function $f \in \mathcal{C}(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R} \setminus \{ 0 \})$ such that
$$ \exists \lim_{x \to 0} f'(x) = \lim_{x \to 0} \lim_{y \to x} \frac{f(y)-f(...
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