What is a continuous function?

What does it mean for a function to be continuous? Here's a (relatively) simple definition of continuity.

The epsilon delta definition of continuity is one of the first you are likely to meet at college mathematics level. This video explains the thought process behind this definition using a simple graphical example.

[didarislam]

Greatly explained! Visualization has always been key to understanding for me. Thank you!

[toolhog]

This is that section in calc 1 way back for limit definitions that I just sorta skipped. Seems pretty straight forward here as you explain it.

[jakeaus]

Would you say that f(x) = log(x) is a continuous function? What about g(x) = sqrt(x)? Or h(x) = 1/x?

In each case the domain is a certain subset of the real numbers, and so there are values for which the function is not defined. But, for every value for which the function *is* defined, the function is also continuous at that value by the epsilon-delta definition given here.

So if a function is continuous at every value in its domain, is it a continuous function? Or is a function only a continuous function if it is locally continuous for every real number input, regardless of whether the function can be evaluated at every real number?

Put another way, we can ask whether f(x) = log(x) is continuous at x = 0. It seems to me that the correct response would be one of either:
i) Yes, f is continuous at x = 0.
ii) No, f is not continuous at x = 0.
iii) f is not defined at x = 0, so it is meaningless to ask whether it is continuous there.
What do you think?

[ultimateo]

What if you graph in a higher dimension. You then could go from a position on the x axis to another by moving through a different z position.

Would this explanation work with any x number of players in any x number of dimensions

[yapibowler]

Sir! You have a great explanation. But you seem to have very fast discussion. I honestly can't cope with the speed of your discussion sir. Hope to make notice on this comment sir, thank you!

[juliolobo]

very nice one but you actually talk to fast..while talking you should use cursor to point to the part that you are explaining..quite a bit hard to follow