continuity

[violet adder]

if I've been given a function, how do I find the number of points it is discontinuous at?

[torn shoal]

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[shadow valve]

discontinuity of of different types

removable discontinuity is when there is a hole in graph

like for example (x+1)/(x+1)

its obviously 1 not at x=-1

[violet adder]

yeah

[shadow valve]

its like a hole in the graph

similarly sinx/x

[violet adder]

at 0

[shadow valve]

yup but the catch is limit at that point exist

some cases limits dont exist

like in 1/x

we need to look for assymptotes

or maybe i am giving very simple examples

[violet adder]

yeah what about cases with multiple points of discontinuity

[shadow valve]

some times discontinuity is like a jump in graph like in step x

if there is a problem i might explain better

[shadow valve]

in such questions i tend making graphs

(-1,1) is a small interval

and finding where 2x^2+1 is integers coz theres where discontinuity is

[violet adder]

the interval is (-1,1)

oh wait

[shadow valve]

$f \circ g (x)= [2(2x-3)^2+1]$
for x <0

[hybrid bear]

pratham

[shadow valve]

now this thing is discontinuous when the thing inside step is an integer ryt?

[violet adder]

yeah

[shadow valve]

now i can see that at x=-1/2 its a integer

so its not continuous at that point

at '0' check left hand and right hand limit
if lhl and rhl are unequal its discontinuouus at that point

[violet adder]

its equal

[shadow valve]

if its equal the function is continuous

i hope this approach helps

[violet adder]

what's the answer then?

[shadow valve]

find out

[cloud estuary]

is [t] like a stairs function ?

[shadow valve]

$f \circ g (x)= [2(2x+3)^2+1]$
for $x \geq 0$

[hybrid bear]

pratham

[shadow valve]

floor function yes

[violet adder]

at 1/2

so 2 points?

[shadow valve]

no wait

well @violet adder this approach isnt efficient

there are so many points where the thing inside is integer

[cloud estuary]

wow

[shadow valve]

yup

say x=1
2x+3=5
now 2(5^2)+1=51 ryt?

[cloud estuary]

that not included

interval

[shadow valve]

yup thats a limit kinda thing ryt

say x=0
fog=19

so in (0,1) we might get discontinuity at
y=19,20,21,....,50 (coz 51 not incuded)

thats for x >= 0

i hope that makes sense

[cloud estuary]

no....

y gets only 3 values I think

maybe

[shadow valve]

why?

[cloud estuary]

cause of that f[t] definition

or not

lol my mistake

right

is 19 the lowest?

[shadow valve]

yup

coz (0,1)

now we can find for (-1,0) too

[cloud estuary]

nice

same number both sides?

[shadow valve]

maybe

its better tio check same way..

[cloud estuary]

oh maybe same values

so in total 32 haha

[violet adder]

so 32+31?

[cloud estuary]

but f°g[-1/2]= f°g[1/2] right?

[violet adder]

but 0 is only in one of the cases

[cloud estuary]

oh