First, did you graph the function near zero?
Had you, you would have seen that the graph osculates wildly between near zero.
This is such a famous graph it is known as the topologist sine curve.
So no limit exist at .
Let:
Show that f is not continuous for x = 0
For both and we have that they are a number [-1,1]. We also see that is not defined.
How to proceed?
What I know is that if , then the function is continuous. But what role does the have?
Plato is referring to the sequential definition of continuity - Continuous function - Wikipedia, the free encyclopedia
Quite the contrary! f(0) is defined- it is 0.
If it were true that f(0) were undefined, then you would be done. the limit cannot equal f(0) if f(0) is undefined! But it is defined so in order not to be continuous, either the limit does not exist or the limit exists but is not equal to 0.How to proceed?
What I know is that if , then the function is continuous. But what role does the have?
What is ? You asset that the two one sides limits "are a number in [-1, 1]" but you don't show that. What are those limits? In order that the limit exist the two one-sided limits must be the same.