From the course: Complete Guide to Calculus Foundations for Data Science
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Continuity with derivatives
From the course: Complete Guide to Calculus Foundations for Data Science
Continuity with derivatives
- [Instructor] Next up, let's talk about continuity derivatives and how these two topics are related. You learned about continuity in the previous chapter, so let's take a moment to review what a continuous function is. Remember, a function f is continuous at a point x equals a where the following three conditions are true. One, the function f is defined at the point x equals a. Two, the function has a limit that exists as x approaches a. Three, the limit as x approaches a is f of x equals f of a. If this is not the case, then the function is discontinuous at that point. Also remember that a function f of x is continuous on an open interval a to b if it is continuous for all points within that interval. Now, let's take a look at what differentiable means. A function f of x is differentiable at a point x equals a if the derivative f prime of a exists. To expand upon that, a function f of x is differentiable on an interval a to b if the derivative f prime of a exists for every point on…
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