How can I know if a function is continuous without looking at its graph?

[abel muroi]

I was given this problem..

y = (cos x)/(5x)


and i was asked to determine at which points is this function continuous?

I know I can just graph the function and find the points from there, but is there another method for finding all the points in which the function is continuous? (sketching the graph during a test is very time consuming..)

 

[Ishuda]

I was given this problem..

y = (cos x)/(5x)


and i was asked to determine at which points is this function continuous?

I know I can just graph the function and find the points from there, but is there another method for finding all the points in which the function is continuous? (sketching the graph during a test is very time consuming..)

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

Hint: Look for places where the laws of mathematics are broken. For example what happens when x is 3 in the formula
y = $\displaystyle \frac{12}{x-3}$

 

[pka]

I was given this problem..
y = (cos x)/(5x)
and i was asked to determine at which points is this function continuous?

When you began studying functions, you were asked about domains.
The given function is defined for all non-zero real numbers. Therefor zero is a problem.
The function is not continuous for x=0. Experience should teach you that both cos(x) & 5x are both well behaved. Hence to answer here is (cos(x))/(5x) is continuous everywhere except at zero.

The most important word in that discussion is experience. You should make ever effort to learn basic classes of continuous functions. Understand the results of their compositions.

 

[abel muroi]

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

Hint: Look for places where the laws of mathematics are broken. For example what happens when x is 3 in the formula
y = $\displaystyle \frac{12}{x-3}$

Well, I usually try to remember the domains of trig functions.

Since the domain of cos x is all real numbers, that means that the function (cos x)/(5x) will be continuous everywhere EXCEPT 0, (since a fraction that has 0 in the denominator is undefined).

So i think the answer is (-infinity, 0) u (0, +infinity)

is this the correct method for finding out the points where a function is continuous?

 

[stapel]

I was given this [function:] y = (cos x)/(5x)

and i was asked to determine at which points is this function continuous?

I know I can just graph the function and find the points from there, but is there another method for finding all the points in which the function is continuous?

Yes: Don't look for "all the points [at] which the function is continuous"; look for the (many times fewer) points at which the function is not continuous. Do the basic checking: zeroes in any denominator? negatives inside any square roots? Then remember the other checking: non-positives inside a logarithm? out-of-bounds values with inverse-trig functions?

Mostly, that's pretty much it. Of course, polynomials are continuous everywhere, as are sines and cosines (assuming their arguments aren't funky). But mostly it comes down to practice, so you get comfortable with quickly "seeing" the important and useful stuff.