14.1 Continuous functions[2DP]
[2DN] Let \(A⊆ ℝ\) and \(f:A→ℝ\) be a function; let \(x\in A\); \(f\) is called continuous at \(x\) if
\(f\) is called continuous if it is continuous in every point.
The set of all continuous functions \(f:A→ℝ\) is denoted with \(C(A)\); it is a vector space.
Further informations on this subject may be found in Chap. 3 in [ 4 ] , or Chap. 4 of [ 22 ] .
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[14K]Suppose that \(f:(0,1]→ℝ\) is a continuous function. Prove that, it is bounded from above 1 if and only if \(\limsup _{x→ 0+}f(x){\lt}+∞\).
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[14M]Prerequisites:2.Let \(f:ℝ→ℝ\) be a bounded function. Let it be shown that there at most countably many points where a discontinuity may be removed (i.e. the points \(z\) for which \(\lim _{x→ z} f(x) ≠ f(z)\), see [ 52 ] ).
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[14N]Prerequisites:2.Let \(f:ℝ→ℝ\) be a bounded function. Show that the set of discontinuity points of the second type is countable at most (i.e. the points \(z\) where the lateral limits exist but \(\lim _{x→ z+} f(x) ≠ \lim _{x→ z-} f(x)\), see [ 52 ] ).
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[21N]Prerequisites:3.Fixed \(𝛼 {\gt}1\) we define, for \(x∈ℝ\), \(𝛼^ x\) as in 3. Show that this is a continuous function and that it is a homeomorphism between \(ℝ\) and \((0,∞)\). The inverse of \(y=𝛼^ x\) is the function logarithm \(x=\log _𝛼 y\).
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[14P]Prerequisites:4.Difficulty:*.
Let \(C⊂ℝ\) be a closed set, and let \(f:C→ ℝ\) be continuous function. Show that there exists \(g:ℝ →ℝ\) continuous and extending \(f\), i.e. \(g_{|_ C}=f\).
Hidden solution: [UNACCESSIBLE UUID ’14Q’]
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[14R]Difficulty:**.Find a continuous function \(f:{\mathbb {R}}\to {\mathbb {R}}\) that is not monotonic in any interval (open nonempty). [UNACCESSIBLE UUID ’14S’]
Given a continuous function \(f=f(x,y):ℝ×[0,1]→ℝ\), setting
show that \(g\) is continuous.
Hidden solution: [UNACCESSIBLE UUID ’14V’] [14W] Given a continuous function \(f=f(x,y):ℝ×[0,1]→ℝ\) , and setting
show that \(g\) is continuous. Hidden solution: [UNACCESSIBLE UUID ’14X’] [14Y] Let \(x_ n,y_ n\) be strictly positive real sequences with limit zero; there is a continuous and monotonic function \(f:[0,∞)→ [0,∞)\) such that \(f(0)=0\) and \(∀ x{\gt}0, f(x){\gt}0\), and such that \(∀ n, f(x_ n){\lt}y_ n\) (hence \(\lim _{x→ 0+}f(x)=0\)).
Hidden solution: [UNACCESSIBLE UUID ’14Z’] [150] Let be given a function \(g:[0,∞)→ [0,∞]\) such that \(g(0)=0\) and \(\lim _{x→ 0+}g(x)=0\); then there exists a continuous and monotonic function \(f:[0,∞)→ [0,∞]\) such that \(f(0)=0\), \(\lim _{x→ 0+}f(x)=0\), and \(f≥ g\). [151]Prove that if a monotonic function is defined on a dense subset of an open interval \(I\), and has dense image in another open interval \(J\), then it can be extended to a monotonic continuous function between the two open intervals \(I,J\).
(What happens if \(I\) is closed but \(J\) is open?) [152]Prerequisites:categories of Baire Sec. 10.11.Difficulty:*.
Show that there is no function \(f:ℝ→ℝ\) which is continuous on the rational points and discontinuous on the irrational points. (Hint. Show that the set \(ℝ⧵ℚ\) of irrationals is not a \(F_𝜎\) set in \(ℝ\), using Baire’s theorem.)
Hidden solution: [UNACCESSIBLE UUID ’153’]