Question

Discuss the continuity of the function f, where f is defined by

f(x) =

3, if 0< or = x < or = 1
4, if 1 < x < 3
5, if 3, < or = x < or = 10

@rational

Answer 1

Clearly the function is jumping at x=1 and x=3 and continuous in between.

You may start with that observation and elaborate on that

Answer 2

drawing tool is acting up :/

Answer 3

Should I take limit of 1 from negative and positive?

make it out as

Limit of x -> 1^- f(x) = limit of x -> 1^- (3) = 3
and for positive, it becomes 4

f(1) = 3
LHL =/=RHL =/= f(1)

So it is not continuous?

Answer 4

you may use the graph to help justifying stuff
http://gyazo.com/39bbbd19393a6a4c686e040ddc0da936

Answer 5

How did you decide it is 1 and 3 and not 1 and 10 or 3 and 10?

Answer 6

look at the graph
first convince yourself that it is continuous everywhere except at x=1 and x=3

Answer 7

you may use LHL/RHL to justify.. but first get a feel of the graph

Answer 8

How did you graph this? It would be helpful to do it in the side when I'm doing this

Answer 9

It is easy to sketch it from the given piecewise definition

```
3, if 0< or = x < or = 1
```

the function stays at 3, between x=0 and x=1

Answer 10

or are you asking how to graph it using tools like desmos/wolfram ?

Answer 11

no no I mean just graph xD

I got it, since they did not define anything for 10 we take only 1 and 3, am I right?

Answer 12

Yes we care about continuity only in the domain of definition of function

Answer 13

x<0 and x>10 are not part of domain

Answer 14

Thanks I totally got it.

Off-topic: one of my teachers taught my friend on solving something in a mathematical way, with the graphs, why this happens and what not, my friend stared at him confused, so he said "dude just put lambda here, get the value, and your done" and my friend was like ooooh XD

Answer 15

graphs should not be used in proofs
graphs allow you to understand the problem/concept better

Answer 16

well he was learning a few days before exam so he just wanted to write the answer, not know why the answer is that way xD

Answer 17

thats okay :) you cannot show a graph and say "the relative maximum is lying right at x=3" etc..

Answer 18

that wont work in exams atleast..

Answer 19

I'll try my best not to learn it along with the concepts, I have about 4 chapters left out[half done] so I'm trying to hurry, complete it by 16th so I can sleep well before the exam

Answer 20

good luck with the preparation :)

Answer 21

Thank you(: I owe you big time @rational