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HallsofIvy said:Since, in the statement of the problem, f is only required to be continuous, not differentiable, what does f ' mean?
wdlang said:f is continuous means f is infinitely differentiable
Whovian said:No it doesn't. Assume ##\displaystyle f\left(x\right)=\left|x\right|##. It's obviously continuous (##\displaystyle\forall a\in\mathbb{R}\ \left(\lim_{x\to a}\left|x\right|=\left|a\right|\right)##), but its derivative at 0 doesn't exist.
wdlang said:ok, i mean smooth
sorry for that
Whovian said:Okay. ##\displaystyle\dfrac{x\cdot\left|x\right|}{2}##. It's smooth, but its derivative is |x|.
Though I'm pretty sure you mean continuously differentiable, which is ideally what we'd assume.
A functional analysis problem is a type of mathematical problem that involves finding the optimal function that satisfies a given set of constraints. This type of problem is commonly encountered in fields such as economics, engineering, and physics.
The first step in solving a functional analysis problem is to define the problem and identify the relevant variables. Next, mathematical models are created and various techniques such as calculus and optimization are used to analyze the problem. Finally, a solution is found by identifying the function that maximizes or minimizes the given objective function while satisfying all constraints.
Functional analysis problems have a wide range of applications in various fields such as economics, engineering, physics, and biology. They are used to analyze and optimize systems, predict outcomes, and make decisions.
Some common techniques used to solve functional analysis problems include calculus, linear algebra, convex analysis, and optimization methods such as gradient descent, Lagrange multipliers, and dynamic programming.
Functional analysis problems can become very complex and difficult to solve for real-world systems with many variables and constraints. In some cases, the optimal solution may not be practical or feasible in the real world due to various limitations and uncertainties.