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Problem 59

Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+3 \mathrm{y}\) and let \(\varepsilon>0\) be given. Prove that \(\mathrm{F}\) is continuous in the whole plane by finding \(\delta>0\) such that for $\left|(x, y)-\left(x_{0}, y_{0}\right)\right|<\delta$ \(\left|F(x, y)-F\left(x_{0}, y_{0}\right)\right|<\varepsilon\) where $x_{0}, y_{0}$ is an arbitrary point in the plane.

Expert verified

For the given function \(F(x, y)=x^2+3y\), we can find a bound \(\delta>0\) such that if the distance between \((x, y)\) and \((x_0, y_0)\) is less than \(\delta\), then the distance between \(F(x, y)\) and \(F(x_0, y_0)\) is less than \(\varepsilon\). We have found that \(\delta = \frac{\varepsilon}{2 (|x_0| + 1) + 7}\) satisfies this condition. Therefore, the function \(F(x, y)=x^2+3y\) is continuous in the whole plane.

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Chapter 3

Show that the exponential function defined by $$ \exp \mathrm{x}=\lim _{\mathrm{n} \rightarrow \infty}[1+(\mathrm{x} / \mathrm{n})]^{\mathrm{n}} $$ is everywhere continuous.

Chapter 3

Which of the following functions are uniformly continuous on the specified domains? a) \(f(x)=x^{3}(0 \leq x \leq 1)\) b) \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}(0 \leq \mathrm{x}<\infty)\) c) \(f(x)=\sin x^{2}(0 \leq x<\infty)\) d) \(f(x)=1 /\left(1+x^{2}\right)(0 \leq x<\infty)\)

Chapter 3

Show why the function, $\mathrm{f}(\mathrm{x})={ }^{\infty} \Sigma_{\mathrm{n}=0} \mathrm{a}^{\mathrm{n}} \cos \left(b^{\mathrm{n}} \pi \mathrm{x}\right) \quad 0<\mathrm{a}<1 \quad \mathrm{~b}=Z \mathrm{k}-1$ constructed by Weierstrass, is continuous everywhere but differentiable nowhere.

Chapter 3

Let \(\mathrm{f}: \mathrm{R}^{3} \rightarrow \mathrm{R}\) be given by $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left[\left(\mathrm{y}^{3} \mathrm{z}\right) /\left(1+\mathrm{x}^{2}+\mathrm{z}^{3}\right)\right]$ ] Show that \(\mathrm{f}\) is continuous at \((0,0,0)\).

Chapter 3

What is meant by a discontinuity of the first kind? A discontinuity of the second kind? Show that monotonic functions have no discontinuities of the second kind.

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