Prove that f is continuous at 𝐗0 : (a) f(x, y, z)=x^2+2 y^2-z, 𝐗0=(a, b, c). (b) f(x, y)=cos(x

step 1 We want to prove that this function is continuous at 0 .0. So for it to be continuous, we want t

Prove that f is continuous at 𝐗0 : (a) f(x, y, z)=x^2+2...

Prove that $f$ is continuous at $\mathbf{X}_0$ : (a) $f(x, y, z)=x^2+2 y^2-z, \quad \mathbf{X}_0=(a, b, c)$. (b) $f(x, y)=\cos \left(x^2+y^2\right), \quad \mathbf{X}_0=\left(\sqrt{\frac{\pi}{2}}, \sqrt{\pi}\right)$. (c) $\quad f(x, y, z)=\exp \left(x^2+y+z^2\right), \quad \mathbf{X}_0=(1,0,1)$. (d) $f(x, y)=\log \left(x^2+y^2\right), \quad \mathbf{X}_0=(1,-1)$.

Prove that $f$ is continuous at $\mathbf{X}_0$ :
(a) $f(x, y, z)=x^2+2 y^2-z, \quad \mathbf{X}_0=(a, b, c)$.
(b) $f(x, y)=\cos \left(x^2+y^2\right), \quad \mathbf{X}_0=\left(\sqrt{\frac{\pi}{2}}, \sqrt{\pi}\right)$.
(c) $\quad f(x, y, z)=\exp \left(x^2+y+z^2\right), \quad \mathbf{X}_0=(1,0,1)$.
(d) $f(x, y)=\log \left(x^2+y^2\right), \quad \mathbf{X}_0=(1,-1)$.

Key Concepts

Continuity of Multivariable Functions

Continuity in several variables means that a function f is continuous at a point X? if the limit of f(X) as X approaches X? is equal to f(X?). This generalizes the one-variable concept by considering limits along all possible paths in ?? and is essential for ensuring that small changes in input lead to small changes in output.

Polynomial Functions

Polynomial functions in multiple variables are constructed from sums and products of the variables raised to nonnegative integer powers. They are continuous everywhere in ?? because the operations of addition and multiplication preserve continuity, simplifying the analysis of their behavior at any point in their domain.

Composite Functions

The continuity of composite functions is determined by combining the continuity of the outer function with that of the inner function. If the inner function is continuous at a point and the outer function is continuous at the image of that point, then the composite function is continuous, a fact that is widely used in verifying continuity of functions defined as compositions.

Exponential Functions

Exponential functions, such as exp(x), are continuous for all real numbers. When an exponential function is composed with continuously varying multivariable expressions, the overall function remains continuous, which is a crucial property when dealing with functions in analysis and differential equations.

Logarithmic Functions

Logarithmic functions are continuous on their domains, which consist of the positive real numbers. In multivariable contexts, ensuring that the argument of a logarithmic function stays positive throughout its domain is key for establishing continuity. This concept is central when analyzing functions that involve logarithms.

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