Math Problems Vault

Problem 809

Prove that the only continuous function $f(x)$ ( $-\infty <x<+\infty$ ) satisfying the equation

f(x+y)=f(x)+f(y) \tag {1}

for all real values of $x$ and $y$ is the homogeneous linear function:

f(x)=ax ,

where $a=f(1)$ is an arbitrary constant.

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Problem 810

Prove that a monotone function $f(x)$ satisfying equation (1) is homogeneous linear.

Equation (1): $f(x+y)=f(x)+f(y)$ .

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Problem 811

Prove that a function $f(x)$ satisfying equation (1) and bounded on some small interval $(-\varepsilon ,\varepsilon )$ is a homogeneous linear function.

Equation (1): $f(x+y)=f(x)+f(y)$ .

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Problem 812

Prove that the only continuous function $f(x)$ ( $-\infty <x<+\infty$ ), not identically zero, satisfying the equation

f(x+y)=f(x)f(y) \tag {2}

for all values of $x$ and $y$ is the exponential function:

f(x)=a^{x} ,

where $a=f(1)$ is a positive constant.

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Problem 813

Prove that a function $f(x)$ , not identically zero, bounded on the interval $(0,\varepsilon )$ and satisfying equation (2), is an exponential function.

Equation (2): $f(x+y)=f(x)f(y)$ .

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Problem 814

Prove that the only continuous function $f(x)$ ( $0<x<+\infty$ ), not identically zero, satisfying the equation

f(xy)=f(x)+f(y)

for all positive values of $x$ and $y$ is the logarithmic function:

f(x)=\log _{a}x ,

where $a$ is a positive constant.

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Problem 815

Prove that the only continuous function $f(x)$ ( $0<x<+\infty$ ), not identically zero, satisfying the equation

f(xy)=f(x)f(y) \tag {3}

for all positive values of $x$ and $y$ is the power function:

f(x)=x^{a} ,

where $a$ is a constant.

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Problem 816

Find all continuous functions $f(x)$ ( $-\infty <x<+\infty$ ) satisfying equation (3) for all real values of $x$ and $y$ .

Equation (3): $f(xy)=f(x)f(y)$ .

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Problem 817

Prove that the discontinuous function

f(x)=\operatorname {sgn}x

satisfies equation (3).

Equation (3): $f(xy)=f(x)f(y)$ .

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Problem 818

Find all continuous functions $f(x)$ ( $-\infty <x<+\infty$ ) satisfying, for all real values of $x$ and $y$ , the equation

f(x+y)+f(x-y)=2f(x)f(y)

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Problem 819

Find all bounded continuous functions $f(x)$ and $g(x)$ ( $-\infty <x<+\infty$ ) satisfying, for all real values of $x$ and $y$ , the system of equations

f(x+y)=f(x)f(y)-g(x)g(y)

g(x+y)=f(x)g(y)+f(y)g(x)

together with

f(0)=1 \text{ and } g(0)=0

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Problem 820

Let

\triangle f(x)=f(x+\triangle x)-f(x)

and

\triangle ^{2}f(x)=\triangle \left\{ \triangle f(x)\right\}

be, respectively, the first- and second-order finite differences of the function $f(x)$ .

Prove that if the function $f(x)$ ( $-\infty <x<+\infty$ ) is continuous and

\triangle ^{2}f(x)\equiv 0 ,

then this function is linear, i.e.,

f(x)=ax+b ,

where $a$ and $b$ are constants.

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