Recipe of Continuous Functions
A function 𝑅→𝑅 is continuous at a point 𝑥0,if for 𝜀>0, there exist a 𝛿>0 such that |𝑥−𝑥𝑜|<𝛿 implies
|𝑓(𝑥)−𝑓( 𝑥𝑜)|<𝜀. Therefore, the function f is a continuous function if it is continuous at every point. In the context of general topology, the ‘small’ sets may be replaced by ‘well-chosen open set’. So the definition should be thought of as: a function 𝑓 is continuous at 𝑥 if given any small enough open set U containing 𝑓(𝑥), there is an open set V (that we may be forced to choose very small) containing 𝑥 such that 𝑓(𝑉)⊂𝑈. Continuity of a function depends not only upon the function 𝑓 itself, but also on the topologies specified for its domain and range.