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Omni-MATH / Prove or disprove the following statements: (a) There exists a monotone function…

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problem

Prove or disprove the following statements: (a) There exists a monotone function f:[0,1]→[0,1]f:[0,1] \rightarrow[0,1] such that for each y∈[0,1]y \in[0,1] the equation f(x)=yf(x)=y has uncountably many solutions xx. (b) There exists a continuously differentiable function f:[0,1]→[0,1]f:[0,1] \rightarrow[0,1] such that for each y∈[0,1]y \in[0,1] the equation f(x)=yf(x)=y has uncountably many solutions xx.
Plain-text mathematical notation (without MathML)
Prove or disprove the following statements: (a) There exists a monotone function f:[0,1]→[0,1] such that for each y∈[0,1] the equation f(x)=y has uncountably many solutions x. (b) There exists a continuously differentiable function f:[0,1]→[0,1] such that for each y∈[0,1] the equation f(x)=y has uncountably many solutions x.
Original LaTeX notation
Prove or disprove the following statements: (a) There exists a monotone function $f:[0,1] \rightarrow[0,1]$ such that for each $y \in[0,1]$ the equation $f(x)=y$ has uncountably many solutions $x$. (b) There exists a continuously differentiable function $f:[0,1] \rightarrow[0,1]$ such that for each $y \in[0,1]$ the equation $f(x)=y$ has uncountably many solutions $x$.

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Source and history

task_id
c49cb3d0-d9c4-5d6a-a1fe-a0dae9fbcd05
task_key
test--c49cb3d0-d9c4-5d6a-a1fe-a0dae9fbcd05
task_revision_id
1
config / split
/ test

Official source

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