MHB Implicit Function: What is It and Why Is It Used?

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The discussion centers on the concept of implicit functions and their definition as functions. It highlights the implicit function theorem as a key aspect of understanding implicit functions. Various interpretations and explanations of implicit functions are acknowledged, indicating the complexity of the topic. The conversation suggests that further exploration of the implicit function theorem could enhance understanding. Overall, implicit functions are recognized for their significance in calculus and mathematical analysis.
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Why implicit function is a function?
 
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https://mathhelpboards.com/calculus-10/implicit-function-24656.html are some answers to the same question.
Given such a quite broad question, many different answers are possible.
My personal (Pavlovian) reaction would probably mention the implicit function theorem, but I think the discussion is best continued in that other topic.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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