True or False:

All even functions are continuous everywhere.

and

If -1 (is less than or equal to) g(x) (is less than or equal to) for all x, then limit as approaches infinity of g(x)=0

I think the first one may be false and the second one is false too

Question

True or False: 

All even functions are continuous everywhere. 

and

If -1 (is less than or equal to) g(x) (is less than or equal to) for all x, then limit as approaches infinity of g(x)=0

I think the first one may be false and the second one is false too

dape · Answer

You are right, but the second one didn't really make much sense as you wrote it down. An example of an even function that is discontinuous is this one:
|dw:1377569688783:dw|

dape · Answer

You can probably come up with more examples.

anonymous · Answer

i think that is not a function

anonymous · Answer

|dw:1377569968254:dw|

dape · Answer

You're right, I have looked at too many computer plots.

anonymous · Answer

The second one doesn't make much sense since you've missed out a bit:
$$-1 \le g(x) \leq \ ???$$Not that it really matters. I'm fairly sure it's false. You can always pick g(x) = -1, which will always converge to -1, not 0.

(As a technically, we're assuming the upper bound allows such a function $g$ to exist...)

anonymous · Answer

Not sure what you mean by I have missed a bit. So is the first one true and the secon one false?

anonymous · Answer

The first one is false: there's a counterexample above. 

The second one: you said, "If -1 (is less than or equal to) g(x) (is less than or equal to) for all x". But g(x) (is less than or equal to) what?

Although I did point out that it didn't really matter. The second one is also false.

anonymous · Answer

That example isnt a function though since it doesnt meet the vert line test. I typed exactly what was on the question. For all x

anonymous · Answer

You cannot say "g(x) is less than or equal to for all x". Remember maths should be written in English - if you read a sentence and it makes no sense in English, it probably makes no sense. You need to say what g(x) is less than for all x.

Perhaps you mean $g(x) \leq x$ for all $x$?

As for the first question, satellite removed the offending vertical lines.
|dw:1377575247783:dw|
So this is a valid even and discontinuous function. (The black dots represent a point that is included in the function, while the white dots represent a point not included. You can see this now satisfies the vertical line test.)