T-O7 said:Hey, I have a quick question: Does anyone know what it means for a function [tex]f[/tex] to be continuous on a curve [tex]\gamma[/tex]? Is it the same as saying the pullback [tex]\gamma^*f[/tex] is continuous on the domain of [tex]\gamma[/tex]?
A continuous curve is a smooth, unbroken line that does not have any abrupt changes or breaks. In mathematics, it is a function that is defined and has a value at every point along the curve.
Continuity on a curve is different from continuity on a straight line because on a curve, the function is defined and has a value at every point, including the points where the curve changes direction. On a straight line, the function is defined and has a value at every point along the line, but the line does not change direction, so there are no points where the function is undefined.
Continuity on a curve is important in mathematics because it allows us to accurately represent real-world phenomena and make predictions based on mathematical models. It also enables us to use calculus to find derivatives and integrals, which are essential in solving many mathematical problems.
Continuity on a curve is a necessary condition for differentiability. A function must be continuous at a point in order for it to be differentiable at that point. This means that if a curve is not continuous, it cannot be differentiable.
Yes, a curve can be continuous but not differentiable. This occurs when there is a sharp change or corner in the curve, which means that the derivative of the function does not exist at that point. However, the function can still be continuous at that point if the left and right hand limits of the function are equal.