Uniform Continuity: Example of f*g Not Being Uniformly Continuous

CarmineCortez
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Homework Statement



If f and g are uniformly continuous on X, give an example showing f*g may not be uniformly continuous.



The Attempt at a Solution



i think if the functions are unbounded the product will not uniformly continuous. Is there a specific example of this function..?
 
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Consider some simple examples first!
Choose X = \mathbb{R}, and maybe let f(x) = x, for any x \in \mathbb{R}.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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