Oszillation bzgl. Lebesgue-Kriterium
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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Exercise:
Sei f:Xrightarrow mathbbR eine beschränkte Funktion auf einem metrischen Raum X. Für jedes eta geq ist die Teilmenge N_eta xin X mid omegafxgeq eta subseteq X abgeschlossen.
Solution:
Sei eta geq und sei x_k_k eine konvergente Folge in X mit omegafx_kgeq eta für alle kin mathbbN und mit Grenzwert xin X. Sei delta beliebig. Dann existiert ein k mit x_kin B_deltax. Da B_deltax offen ist gibt es des Weiteren ein delta_k mit B_delta_kx_k subseteq B_deltax. Nun ist textsupfB_deltax &geq textsupfB_delta_kx_k textinffB_deltax &geq textinffB_delta_kx_k und daher omegafxdelta geq omegafx_kdelta_k. Da aber omegafx_kdelta_k geq omegafx_k geq eta gilt erhält man omegafxdelta geq eta. Da delta beliebig war folgt das Lemma.
Sei f:Xrightarrow mathbbR eine beschränkte Funktion auf einem metrischen Raum X. Für jedes eta geq ist die Teilmenge N_eta xin X mid omegafxgeq eta subseteq X abgeschlossen.
Solution:
Sei eta geq und sei x_k_k eine konvergente Folge in X mit omegafx_kgeq eta für alle kin mathbbN und mit Grenzwert xin X. Sei delta beliebig. Dann existiert ein k mit x_kin B_deltax. Da B_deltax offen ist gibt es des Weiteren ein delta_k mit B_delta_kx_k subseteq B_deltax. Nun ist textsupfB_deltax &geq textsupfB_delta_kx_k textinffB_deltax &geq textinffB_delta_kx_k und daher omegafxdelta geq omegafx_kdelta_k. Da aber omegafx_kdelta_k geq omegafx_k geq eta gilt erhält man omegafxdelta geq eta. Da delta beliebig war folgt das Lemma.
Meta Information
Exercise:
Sei f:Xrightarrow mathbbR eine beschränkte Funktion auf einem metrischen Raum X. Für jedes eta geq ist die Teilmenge N_eta xin X mid omegafxgeq eta subseteq X abgeschlossen.
Solution:
Sei eta geq und sei x_k_k eine konvergente Folge in X mit omegafx_kgeq eta für alle kin mathbbN und mit Grenzwert xin X. Sei delta beliebig. Dann existiert ein k mit x_kin B_deltax. Da B_deltax offen ist gibt es des Weiteren ein delta_k mit B_delta_kx_k subseteq B_deltax. Nun ist textsupfB_deltax &geq textsupfB_delta_kx_k textinffB_deltax &geq textinffB_delta_kx_k und daher omegafxdelta geq omegafx_kdelta_k. Da aber omegafx_kdelta_k geq omegafx_k geq eta gilt erhält man omegafxdelta geq eta. Da delta beliebig war folgt das Lemma.
Sei f:Xrightarrow mathbbR eine beschränkte Funktion auf einem metrischen Raum X. Für jedes eta geq ist die Teilmenge N_eta xin X mid omegafxgeq eta subseteq X abgeschlossen.
Solution:
Sei eta geq und sei x_k_k eine konvergente Folge in X mit omegafx_kgeq eta für alle kin mathbbN und mit Grenzwert xin X. Sei delta beliebig. Dann existiert ein k mit x_kin B_deltax. Da B_deltax offen ist gibt es des Weiteren ein delta_k mit B_delta_kx_k subseteq B_deltax. Nun ist textsupfB_deltax &geq textsupfB_delta_kx_k textinffB_deltax &geq textinffB_delta_kx_k und daher omegafxdelta geq omegafx_kdelta_k. Da aber omegafx_kdelta_k geq omegafx_k geq eta gilt erhält man omegafxdelta geq eta. Da delta beliebig war folgt das Lemma.
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