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| bio | website | |
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| location | ||
| age | ||
| visits | member for | 1 year |
| seen | Dec 9 '12 at 22:17 | |
| stats | profile views | 16 |
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Dec 6 |
awarded | Teacher |
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Jul 21 |
answered | Percentages - Find Maximum value. |
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May 20 |
comment |
Differentiablity and Taylor polynomials @Thomas So, what do I need to prove then. Cheers |
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May 20 |
answered | Correlations between 3 random variables |
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May 20 |
asked | Differentiablity and Taylor polynomials |
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May 5 |
comment |
Continuity leads to constant function (Assignment question) Here is another similar type of question proving the function is constant using the continuity definition, Let $f: [-1,1] \to \mathbb{R}$ be a function with $f(x)=f(x^2)$ for all $ x \in (-1,1)$. Suppose that $f$ is continuous at 0. Prove that $f$ is a constant. |
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May 5 |
comment |
Continuity leads to constant function (Assignment question) Thanks all the the great effort made, let's me just recap what we have had so far: $f$ is continuous iff for all $\epsilon >0$, $ x \in B_{\delta}(x_0)$ then $f(x) \in B_{\epsilon}(f(x_0))$ |
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May 5 |
comment |
Continuity leads to constant function (Assignment question) To my understanding, if we can pick a rational point $x_1$ satisfying $|x_1-x_0|< \delta$ such that $|f(x_1)-f(x_0)| \ge \epsilon$ then this will give a contradiction. |
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May 5 |
comment |
Continuity leads to constant function (Assignment question) @Benjamin Lim : This is the first course in Analysis so we have not learnt anything such as Topological spaces and other things, just simply the definition and way to go. Thanks. |
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May 5 |
awarded | Student |
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May 5 |
comment |
Continuity leads to constant function (Assignment question) The formal one as follows For all $ \epsilon >0$, there exists $ \delta >0$ such that for all $ x \in \mathbb{R}$ satisfying $ |x-x_0| < \delta$, one has $ |f(x)-f(x_0)| < \epsilon $. |
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May 5 |
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Continuity leads to constant function (Assignment question) Yes, it was my initial idea of using Density Theorem. However, I do not see the connection between that with the function. |
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May 5 |
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Continuity leads to constant function (Assignment question) Great solution, however, the question requires the answer to use the definition of the continuity. Thereby, I just do not know where to start. |
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May 5 |
asked | Continuity leads to constant function (Assignment question) |
