$f(x)$ is continuous on [0,1]. Every point $(x,f(x)), x\in \Bbb Q$ is on the linear function $y=(f(1)-f(0))x+f(0)$. Prove that every point $(r,f(r))$ (when $r$ is irrational) is on this linear function.
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Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps. – robjohn May 13 '19 at 17:16
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Your problem is essentially equivalent to this one:
Let $f$ and $g$ be continuous functions with $f(x) = g(x)$ for all $x \in \mathbb{Q}$. Show that $f(x) = g(x)$ for all $x \in \mathbb{R}$.
Obviously, the continuity condition is really important. The other property that's used is the density of $\mathbb{Q}$ in $\mathbb{R}$.
Matthew Leingang
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