Lullaby (Calculus)

Hi Our lecture today Actually has us backtrack a little bit To a concept Which is actually More fundamental Than that of Differentiability It's a topic which is called Continuous functions In other words Our lecture today Is concerned with a topic called Continuity And actually The topic of continuity Had its roots Way back At the beginning of our course When we first raised the question Does the limit of f(x) As x approach a Equal f(a) Remember When we first started talking about the limit concept Our intuitive approach was to say look-it As gets x arbitrarily close to a F(x) gets arbitrarily close to f(a) And we saw That this particular definition Had a few Loopholes in it Even though intuitively This is what we would have liked And so What I want to do today Is to see what happens If we can be sure That the limit of f(x) As x approaches a Equals f(a) What is implied If this happens to be A true statement And my first claim is Is that It is Implicitly implied That f(a) Must at least make sense Otherwise we wouldn't have Written it over here