多元函数连续性和方向导数存在性关系的教学探究

Pedagogical Study on the Relationship between Continuity and Existence of Directional Derivatives of a Multivariate Function

多元函数在一点连续和在该点存在方向导数是多元微分学的重要知识点。在数学分析习题课中,引导学生自主探索这两者之间的关系,通过不断尝试、逆向思考、寻求反例等途径,最终发现二者是互不蕴含的关系。探究式教学可以提升学生的发散思维能力,激发探索钻研的主动性,培养不惧挫折、坚持不懈的品质,并帮助学生养成严谨的思维习惯和实事求是的科学态度。

Continuity of a function at one point and existence of directional derivatives at that point are important parts in multivariate differential calculus.In the mathematical analysis exercise class,students are guided to explore the relationship between these two notions,and through continuous attempts,reverse thinking and seeking counter-examples,they can eventually find that each one cannot be implied by the other.Inquiry-based teaching can strengthen students’ability of divergent thinking,mobilize students’initiative to explore and study,cultivate students’quality of perseverance and not fearing setbacks,and let them establish a rigorous thinking habit and a scientific attitude of seeking truth from facts.