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Suppose that š is differentiable and š ā²ā²(š) exists. Prove that š ā²ā²(š) = lim āāš (š(š+ā)ā2š(š)+š(šāā))/ ā^2. Give an example where the above limit exists, but š ā²ā²(š) does not exist.
Show that the function defined as š(š„) = { sin 1 š„ , š„ ā 0 0, š„ = 0 obeys the intermediate value theorem
Let š: (0,1) ā ā be a bounded continuous function. Show that š(š„) = š„(1 ā š„)š(š„) is uniformly continuous.Ā
State suitable conditions and prove that (šš) ā² = šš ā² + š ā²š.
Evaluate
lim 3nĪ£r=1 n^2/(4n+r)^3
nāā
Show whether the following functions are uniformly continuous on the given domain.
1. F(x)=x^3 on [-1,1]
2. F(x)= 2x/2x-1 on [1, infinity]
3. F(x)= sinx/x on (0,1)
4. F(x)= 1/x on (0,1)
1/(3x5)+ā3/(5x8)+ ā5/(7x11) +... test the convergence
Assume that $1<p<+\infty$, a real-valued function $f$ is absolutely continuous on $[a,b]$, and its derivative $f'$ is in $L^p[a,b]$. Prove that $f$ is $\alpha$-Lipschitz, where $\alpha=1/q$, with $q$ being the conjugate exponent of $p$. let f:R-> R be a function. show that the set of points where f is continuous can be written as a countable intersection of open sets
Is there a continuous function f:[0,1]~>[0,1] that is not constant in any nontrivial interval such that f^-1{0} is uncountable?
