Continuous functions (1 Viewer)

[sandersen]

I'm doing some practice questions and one says to investigate whether a function is continuous.

Do we need to know this for 2U on Monday? What does it mean?

 

[PC]

A function is continuous at a particular point if you can find the derivative of the function at that point.

For example, f(x) = 1/x is a discontinuous function. Obviously, since no function value exists for x = 0. When you differentiate you get dy/dx = -1/x² and for that derivative, no value exists when x = 0. But for any other value there is a derivative, so the original function exists for all other points.

Something like that anyway.

 

[sandersen]

I just read that

Descartes said that a function is continuous if its graph can be drawn without lifting the pencil from the paper.

at http://pirate.shu.edu/~wachsmut/ira/cont/contin.html

Does that sound right? Or is there more to it like PC said?

 

[acmilan]

The 'proper' way to find if a function is continuous at a point is to find the limit on both sides at that point and show that equals the function value at that point, ie:

lim(x->a^-) f(x) = lim(x->a⁺) f(x) = f(a)

As PC said, there are other sufficient conditions that imply continuity, such as having a derivative at that point. Its well known that polynomials and functions made of polynomials are continuous, because they are differentiable everywhere.

I honestly doubt they'd make you prove as function is continuous. If the function is made up of polynomials, trig, exponential etc. functions (and any combinations of them) then it can simply be quoted that its continuous.

 

[sandersen]

Well the question was:

f(x) =
x^2 for x<2

2^x for x=2

5-(1/2)x for x>2

Investigate whether f(x) is continuous at x=2

So I could say that as this function has a derivative at x=2, then it is continuous...?

OK thanks for your help!

 

[PC]

Is that site from the US?

I wouldn't trust anything from the US. Their numeracy ability is even lower than ours.