Does this function have to be constant?
$begingroup$
Suppose that a continuous function $f : [0,1] to mathbb{R}_+$ satisfies the following property for all $x in [0,1]$:
$$
f(x)
= frac{3}{2} fleft(frac{3}{4} xright)
- frac{1}{2} fleft(frac{1}{2} xright).
$$
For example: this is clearly satisfied when $f(x) =c = frac{3}{2}c-frac{1}{2}c$. Are there other functions that satisfy this property?
functional-analysis functions
asked Jan 2 at 18:15
TomHTomH
13213
$endgroup$
add a comment |
$begingroup$
Suppose that a continuous function $f : [0,1] to mathbb{R}_+$ satisfies the following property for all $x in [0,1]$:
$$
f(x)
= frac{3}{2} fleft(frac{3}{4} xright)
- frac{1}{2} fleft(frac{1}{2} xright).
$$
For example: this is clearly satisfied when $f(x) =c = frac{3}{2}c-frac{1}{2}c$. Are there other functions that satisfy this property?
functional-analysis functions
asked Jan 2 at 18:15
TomHTomH
13213
$endgroup$
1
$begingroup$
If we had not required continuity, we could easily find many non-constant such functions. Since we require continuity I'm less certain.
$endgroup$
– Arthur
Jan 2 at 18:53
1
$begingroup$
One such non-continuous example is $f(x)=a/x$ for $x>0$, $f(0)=b$ for any constants $a,b geq 0$.
$endgroup$
– Michael
Jan 2 at 19:21
add a comment |
$begingroup$
Suppose that a continuous function $f : [0,1] to mathbb{R}_+$ satisfies the following property for all $x in [0,1]$:
$$
f(x)
= frac{3}{2} fleft(frac{3}{4} xright)
- frac{1}{2} fleft(frac{1}{2} xright).
$$
For example: this is clearly satisfied when $f(x) =c = frac{3}{2}c-frac{1}{2}c$. Are there other functions that satisfy this property?
functional-analysis functions
asked Jan 2 at 18:15
TomHTomH
13213
$endgroup$
Suppose that a continuous function $f : [0,1] to mathbb{R}_+$ satisfies the following property for all $x in [0,1]$:
$$
f(x)
= frac{3}{2} fleft(frac{3}{4} xright)
- frac{1}{2} fleft(frac{1}{2} xright).
$$
For example: this is clearly satisfied when $f(x) =c = frac{3}{2}c-frac{1}{2}c$. Are there other functions that satisfy this property?
functional-analysis functions
functional-analysis functions
asked Jan 2 at 18:15
TomHTomH
13213
asked Jan 2 at 18:15
TomHTomH
13213
asked Jan 2 at 18:15
TomHTomH
13213
asked Jan 2 at 18:15
TomHTomH
13213
asked Jan 2 at 18:15
TomHTomH
13213
13213
1
$begingroup$
If we had not required continuity, we could easily find many non-constant such functions. Since we require continuity I'm less certain.
$endgroup$
– Arthur
Jan 2 at 18:53
1
$begingroup$
One such non-continuous example is $f(x)=a/x$ for $x>0$, $f(0)=b$ for any constants $a,b geq 0$.
$endgroup$
– Michael
Jan 2 at 19:21
add a comment |
1
$begingroup$
If we had not required continuity, we could easily find many non-constant such functions. Since we require continuity I'm less certain.
$endgroup$
– Arthur
Jan 2 at 18:53
1
$begingroup$
One such non-continuous example is $f(x)=a/x$ for $x>0$, $f(0)=b$ for any constants $a,b geq 0$.
$endgroup$
– Michael
Jan 2 at 19:21
1
1
$begingroup$
If we had not required continuity, we could easily find many non-constant such functions. Since we require continuity I'm less certain.
$endgroup$
– Arthur
Jan 2 at 18:53
$begingroup$
If we had not required continuity, we could easily find many non-constant such functions. Since we require continuity I'm less certain.
$endgroup$
– Arthur
Jan 2 at 18:53
1
1
$begingroup$
One such non-continuous example is $f(x)=a/x$ for $x>0$, $f(0)=b$ for any constants $a,b geq 0$.
$endgroup$
– Michael
Jan 2 at 19:21
$begingroup$
One such non-continuous example is $f(x)=a/x$ for $x>0$, $f(0)=b$ for any constants $a,b geq 0$.
$endgroup$
– Michael
Jan 2 at 19:21
add a comment |
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1
$begingroup$
If we had not required continuity, we could easily find many non-constant such functions. Since we require continuity I'm less certain.
$endgroup$
– Arthur
Jan 2 at 18:53
1
$begingroup$
One such non-continuous example is $f(x)=a/x$ for $x>0$, $f(0)=b$ for any constants $a,b geq 0$.
$endgroup$
– Michael
Jan 2 at 19:21