If a function is differentiable then it’s also continuous. This property is very useful when working with functions, because if we know that a function is differentiable, we immediately know that it’s also continuous. How do you prove continuity and differentiability? If a function f(x) is differentiable at a pointRead More →
A function f is differentiable at x=a whenever f′(a) exists, which means that f has a tangent line at (a,f(a)) and thus f is locally linear at the value x=a. Informally, this means that the function looks like a line when viewed up close at (a,f(a)) and that there isRead More →
The Grand Canyon (Figure below) is a good example of lateral continuity. You can clearly see the same rock layers on opposite sides of the canyon. The matching rock layers were deposited at the same time, so they are the same age. What is Steno’s principle of lateral continuity? Steno’sRead More →
Definition A function f:X → Y from a topological space X to a topological space Y is said to be continuous if f−1(V ) is an open set in X for every open set V in Y , where f−1(V ) ≡ {x ∈ X : f(x) ∈ V }.Read More →