Branches of Implicitly Defined Functions

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Problem 3361 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

Prove that the Dirichlet function, which is discontinuous at every point,

y= \begin{cases} 1, & \text{if } x \text{ is rational;} \\ 0, & \text{if } x \text{ is irrational} \end{cases}

satisfies the equation

y^{2}-y=0 .

Problem 3362 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

Let the function $f(x)$ be defined on the interval $(a,b)$ . Under what conditions does the equation

f(x)y=0

have a unique continuous solution $y=0$ for $a<x<b$ ?

Problem 3363 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

Let the functions $f(x)$ and $g(x)$ be defined and continuous on the interval $(a,b)$ . Under what conditions does the equation

f(x)y=g(x)

have a unique continuous solution on the interval $(a,b)$ ?

Problem 3364 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

(a)(b)(c)

Given the equation

x^{2}+y^{2}=1 \tag {1}

and

y=y(x) \quad (-1 \leqslant x \leqslant 1) \tag {2}

is a single-valued function satisfying equation (1).

(a)

How many single-valued functions (2) satisfy equation (1)?

(b)

How many single-valued continuous functions (2) satisfy equation (1)?

(c)

Given: (a) $y(0)=1$ ; (б) $y(1)=0$ , how many single-valued continuous functions (2) satisfy equation (1)?

Problem 3365 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

Given the equation

x^{2}=y^{2} \tag {1}

and

y=y(x) \quad (-\infty <x<+\infty ) \tag {2}

is a single-valued function satisfying equation (1).

(a)

How many single-valued functions (2) satisfy equation (1)?

(b)

How many single-valued continuous functions (2) satisfy equation (1)?

(c)

How many single-valued differentiable functions (2) satisfy equation (1)?

(d)

Given: (a) $y(1)=1$ ; (б) $y(0)=0$ , how many single-valued continuous functions (2) satisfy equation (1)?

(e)

Given $y(1)=1$ and $\delta$ a sufficiently small number, how many single-valued continuous functions $y=y(x)$ $(1-\delta <x<1+\delta )$ satisfy equation (1)?

Problem 3366 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

(a)(b)(c)(d)

The equation

x^{2}+y^{2}=x^{4}+y^{4}

defines $y$ as a multi-valued function of $x$ . In what domains is this function:

(a)

single-valued;

(b)

two-valued;

(c)

three-valued;

(d)

four-valued?

Find the branch points of this function and its single-valued continuous branches.

Problem 3367 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

Find the branch points and the single-valued continuous branches $y=y(x)$ $(-1 \leqslant x \leqslant 1)$ of the multi-valued function $y$ defined by the equation

(x^{2}+y^{2})^{2}=x^{2}-y^{2}

Problem 3368 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.3 "Differentiation of implicit functions"

(a)(b)

Let the function $f(x)$ be continuous for $a<x<b$ , and let the function $\varphi (y)$ be monotonically increasing and continuous for $c<y<d$ . Under what conditions does the equation

\varphi (y)=f(x)

define a single-valued function

y=\varphi ^{-1}\left[f(x)\right] ?

Study the examples: (a) $\sin y+\operatorname {sh} y=x$ ; (б) $e^{-y}=-\sin^{2}x$ .

(a)

(b)

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.