metalbec said:How do I show that f is differentiable, but f' is discontinuous at 0?
metalbec said:well. f(x)= x^2sin(1/x). I can define f(0) to be 0.
metalbec said:Would I use the Pinching Theorem to show that because the limit as h approaches 0 from both sides is 0, then the limit of hsin (1/h) is also 0?
metalbec said:If I have already established that f'(0) does not exist
Differentiability is a property of a function where it has a well-defined derivative at a given point. Continuity, on the other hand, refers to a function's ability to have a smooth and unbroken graph without any abrupt changes. In simpler terms, differentiability deals with the slope of a function, while continuity deals with the connectedness of its graph.
A function is differentiable at a point if the limit of its derivative exists at that point. This means that the right-hand and left-hand derivatives must be equal at that point, and the function must be continuous at that point as well.
Yes, a function can be continuous but not differentiable at a given point. This happens when the function has a sharp turn or corner at that point, making the derivative undefined. A classic example of this is the absolute value function, which is continuous but not differentiable at x=0.
Differentiability and continuity are essential concepts in calculus and are used to analyze and model real-world phenomena in fields such as physics, engineering, and economics. They allow us to understand and predict the behavior of functions and their rates of change, which are crucial in various applications, such as optimization and motion analysis.
Yes, there are functions that are neither differentiable nor continuous. These are called "discontinuous" functions and can have various forms, such as jump discontinuities, removable discontinuities, and oscillating discontinuities. A classic example is the Dirichlet function, which is continuous at irrational numbers but discontinuous at rational numbers.