3 [0,1] Problem that Justify continuous usfotey that the following rational statement: There exists an increasing function.
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The statement is "There exists an increasing function". Show more…
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3. (a) Show that if a function is continuous on all of R and equal to 0 at every rational number, then the function must be the constant function 0 on all of R. (b) Let f and g be continuous functions on all of R, and f(r) = g(r) for each rational number r. Determine whether or not f(x) = g(x) for all real numbers. (Hint: use (a)).
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Suppose that f : R -> R is a continuous function with the property that f(r) = 0 for every rational number r in R. Prove that f(x) = 0 for all x in R.
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Give a convincing argument that the following function is not continuous at any real number. $$f(x)=\left\{\begin{array}{ll}1, & \text { if } x \text { is rational } \\0, & \text { if } x \text { is irrational }\end{array}\right.$$
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