Convexity conditions and intersections with smooth functions. (English) Zbl 0601.26007
If a continuous real function on an interval I agrees with every straight line in at most finitely many points then there is a subinterval of I on which the function is convex or concave.
If the graph of a continuous function meets every straight line in at most 3 points then the interval I can be decompared into finitely many intervals on each of which the function is convex or concave.
If the number of common points of a function and a straight line is at most 4 then the generalization of the previous theorem is no longer valid.
Those properties and others that generalize them are discussed in the article.
If the graph of a continuous function meets every straight line in at most 3 points then the interval I can be decompared into finitely many intervals on each of which the function is convex or concave.
If the number of common points of a function and a straight line is at most 4 then the generalization of the previous theorem is no longer valid.
Those properties and others that generalize them are discussed in the article.
Reviewer: J.Jedrezejewski
MSC:
| 26A51 | Convexity of real functions in one variable, generalizations |
| 26A48 | Monotonic functions, generalizations |
| 26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
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