e(ho0n3 said:The answer is false: Define f by f(x) = 0 if x is irrational; and f(x) = 1/q where x = p/q, p coprime to q. This function is continuous at every irrational and discontinuous at all the rationals.
Continuity is a property of a function where it is unbroken or uninterrupted over a certain interval or domain. In other words, a function is continuous if there are no abrupt changes or gaps in its graph.
To determine if a function is continuous at a specific point, you need to check if the limit of the function as it approaches that point is equal to the value of the function at that point. If the limit and the function value are the same, then the function is continuous at that point.
The three types of continuity are point continuity, left-sided continuity, and right-sided continuity. Point continuity means that a function is continuous at a specific point. Left-sided continuity means that a function is continuous on the left side of a point. Right-sided continuity means that a function is continuous on the right side of a point.
Yes, a function can be continuous at one point but not at another. This is because continuity is a local property, meaning it only applies to a specific point. A function can be continuous at one point but have a discontinuity at another point.
Differentiability is a stronger condition than continuity. A function is differentiable at a point if it is continuous at that point and has a defined derivative. This means that all differentiable functions are also continuous, but not all continuous functions are differentiable.