A secondary school student's understanding of the concept of function - A case study

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Is the derivative a function? If so, how do we teach it? (2015, background): “, 2008), which often raises difficulties when students try to understand this notation (e.g. Gray & Tall, 1994; Sajka, 2003; Sierpinska, 1994).”
Ideas foundational to calculus learning and their links to students’ difficulties (2021, background): “…with smooth continuous variation when] the person thinks of variation of a variable’s … value as increasing or decreasing … by intervals while 2 Here we finesse the matter of function notation, which is notoriously confusing for students and teachers (Carlson, 1998; Even, 1990; Sajka, 2003).”