Continuity of a function refers to the property of a function where the value of the function at a given point is close to the value of the function at a nearby point. In other words, if the input to the function is close to a particular value, then the output of the function is also close to the corresponding output value.
A continuous function is a function that is continuous at every point in its domain. This means that the function has no sudden jumps, breaks, or holes in its graph.
Continuity and differentiability are closely related properties of a function. A function is differentiable at a point if it is continuous at that point and has a defined derivative at that point. In other words, a function must be continuous in order for it to be differentiable at a particular point.
A discontinuous function is a function that is not continuous at one or more points in its domain. This means that there is at least one point where the function has a break or jump in its graph, or where the function is not defined.
To determine the continuity of a function, you can use the three-part definition of continuity: 1) the function must be defined at the point in question, 2) the limit of the function at that point must exist, and 3) the limit of the function at that point must be equal to the value of the function at that point. If all three conditions are satisfied, then the function is continuous at that point.