I need homework help for calc. I need to create a function that as x approaches point c from left to right?

Question

they are equal. Point c also must be defined, but the function cannot be continuous at point c.

I have really now idea how to do this! Does any have an example?

Answer 1

What are the requirements for continuity of a function f(x) at x=c? They are:

f is defined at c,
the (two-sided) limit lim (x->c) f(x) exists, and
lim(x->c) f(x) = f(c)

Well, you're given that the first two hold. So...

Answer 2

If the function as x approaches c from both left and right approaches the same value, and f(c) is defined, then f is continuous at c, isn't it? But it might not be differentiable if there's a sharp corner at (c,f(c)). The absolute value function gives you that. f(x) = |x-2| is continuous at x = 2, and f(2) is 0, well defined, but it's not differentiable because of the corner.

Unless you mean a piecewise defined function. Something like f(x) = (x²-4)/(x-2) approaches a value of 4 at x=2, but there's a HOLE there, since f(x) is not defined for x=2. But if you separately define f(2), you can make it anything you want. If you make it 4, you've got a removable discontinuity.

On the other hand, something like f(x) = 1/x² can have f(0) = 1 added as a piecewise definition, but since f(x) gets very very large as x→0, there's not a limit, no approaching a single value, and hence no continuity.