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I dont know how to show the functions of this type if it is continuous or not in topological space. I know the method is to use one of two properties: the inverse image of eah open is open, and the inverse image of each closed is closed but i dont know how to use them. i will be thaked if someone help ed me with the steps if f:(R,U) −> (R,U) defined by f(x) = { (((1/3) x +1 , x>3)),(((1/2) (x+5) , x≦3)) :} Is f continuous?

$${I}\:{dont}\:{know}\:{how}\:{to}\:{show}\:{the}\:{functions}\:{of}\: \\ $$$${this}\:{type}\:{if}\:{it}\:{is}\:{continuous}\:{or}\:{not}\:{in}\:{topological}\:{space}. \\ $$$${I}\:{know}\:{the}\:{method}\:{is}\:{to}\:{use}\:{one}\:{of}\:{two}\:{properties}: \\ $$$${the}\:{inverse}\:{image}\:{of}\:{eah}\:{open}\:{is}\:{open}, \\ $$$${and}\:{the}\:{inverse}\:{image}\:{of}\:{each}\:{closed}\:{is}\:{closed} \\ $$$${but}\:{i}\:{dont}\:{know}\:{how}\:{to}\:{use}\:{them}. \\ $$$${i}\:{will}\:{be}\:{thaked}\:{if}\:{someone}\:{help}\:{ed}\:{me}\:{with}\:{the}\:{steps} \\ $$$$ \\ $$$${if}\:{f}:\left(\mathbb{R},{U}\right)\:−>\:\left(\mathbb{R},{U}\right)\:{defined}\:{by} \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:\begin{cases}{\frac{\mathrm{1}}{\mathrm{3}}\:{x}\:+\mathrm{1}\:\:\:,\:{x}>\mathrm{3}}\\{\frac{\mathrm{1}}{\mathrm{2}}\:\left({x}+\mathrm{5}\right)\:\:,\:{x}\leqq\mathrm{3}}\end{cases} \\ $$$${Is}\:{f}\:{continuous}? \\ $$

Question Number 79079 Answers: 1 Comments: 0

Question Number 79075 Answers: 1 Comments: 0

Hello solve in [0;2π] tan2x≥(√3)

$$\mathrm{Hello}\: \\ $$$$\mathrm{solve}\:\mathrm{in}\:\left[\mathrm{0};\mathrm{2}\pi\right]\:\mathrm{tan2}{x}\geqslant\sqrt{\mathrm{3}} \\ $$

Question Number 79065 Answers: 0 Comments: 5

]. lim_(x→0) (([sin(α+β)x+sin(α−β)x+sin2αx)/(cos2βx−cos2αx)).x

$$\left.\right].\:{li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\:\frac{\left[{sin}\left(\alpha+\beta\right){x}+{sin}\left(\alpha−\beta\right){x}+{sin}\mathrm{2}\alpha{x}\right.}{{cos}\mathrm{2}\beta{x}−{cos}\mathrm{2}\alpha{x}}.{x} \\ $$

Question Number 79064 Answers: 1 Comments: 0

find lim_(x→ −3) ((x^3 +27)/(x^5 +243))

$${find}\:\:{li}\underset{{x}\rightarrow\:−\mathrm{3}} {{m}}\frac{{x}^{\mathrm{3}} +\mathrm{27}}{{x}^{\mathrm{5}} +\mathrm{243}} \\ $$

Question Number 79062 Answers: 1 Comments: 0

if a is a rational number with ∣a∣≤1, prove that cos (n cos^(−1) (a)) is also a rational number. (n∈N)

$${if}\:{a}\:{is}\:{a}\:{rational}\:{number}\:{with}\:\mid{a}\mid\leqslant\mathrm{1}, \\ $$$${prove}\:{that}\:\mathrm{cos}\:\left({n}\:\mathrm{cos}^{−\mathrm{1}} \left({a}\right)\right)\:{is}\:{also}\:{a} \\ $$$${rational}\:{number}.\:\left({n}\in\mathbb{N}\right) \\ $$

Question Number 79059 Answers: 0 Comments: 4

∫ cos^2 (x)sin^4 (x) dx ?

$$ \\ $$$$ \\ $$$$\int\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 79051 Answers: 0 Comments: 1

Is Σ_(n=1) ^∞ (1/n) converges or diverges

$${Is}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}\:{converges}\:{or}\:{diverges} \\ $$

Question Number 79049 Answers: 0 Comments: 0

Prove: lim_(x→∞) ((π(x))/(x/ln(x)))=1

$${Prove}: \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\pi\left({x}\right)}{{x}/{ln}\left({x}\right)}=\mathrm{1} \\ $$

Question Number 79055 Answers: 0 Comments: 3

The number of real numbers in [0,2pi] satisfying sin^(−1) x−2sin^2 x+1=0 is

$${The}\:{number}\:{of}\:{real}\:{numbers}\:{in}\:\left[\mathrm{0},\mathrm{2}{pi}\right]\:{satisfying}\:\mathrm{sin}^{−\mathrm{1}} {x}−\mathrm{2}{sin}^{\mathrm{2}} {x}+\mathrm{1}=\mathrm{0}\:{is} \\ $$

Question Number 79052 Answers: 0 Comments: 0

if ∣adj(A)∣=4 of order2,then ∣adj(3A)∣

$${if}\:\mid{adj}\left({A}\right)\mid=\mathrm{4}\:{of}\:{order}\mathrm{2},{then}\:\mid{adj}\left(\mathrm{3}{A}\right)\mid \\ $$

Question Number 79054 Answers: 0 Comments: 0

The angle between the planes r^→ .(2i^→ +2j^→ +2k^→ )=4 and 4x−2y+2z=15 is

$${The}\:{angle}\:{between}\:{the}\:{planes}\:\overset{\rightarrow} {{r}}.\left(\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}}+\mathrm{2}\overset{\rightarrow} {{k}}\right)=\mathrm{4}\:{and}\:\mathrm{4}{x}−\mathrm{2}{y}+\mathrm{2}{z}=\mathrm{15}\:{is} \\ $$

Question Number 79053 Answers: 0 Comments: 0

if (dy/dx) +y=((1+y)/x),then the integrating factor (I,F) is

$${if}\:\:\frac{{dy}}{{dx}}\:+{y}=\frac{\mathrm{1}+{y}}{{x}},{then}\:{the}\:{integrating}\:{factor}\:\left({I},{F}\right)\:{is} \\ $$

Question Number 79026 Answers: 4 Comments: 5

Question Number 79015 Answers: 0 Comments: 4

Rigorously over one month′s time, I developed a formula for general cubic. x^3 +ax^2 +bx+c=0 let x=((pt+q)/(t+1)) pq=m, p+q=s ________________________ m^2 {(a^2 +b)^2 −6a(ab−c)} +m{2(b^2 +ac)(a^2 +b)− 3(ab−c)(ab+3c)} +(b^2 +ac)^2 −6bc(ab−c)=0 ________________________ s=−(2/3){((m(a^2 +b)+b^2 +ac)/(ab−c))} +{(8/(27))[((m(a^2 +b)+b^2 +ac)/(ab−c))]^3 −8[((m^3 +bm^2 +acm+c^2 )/(ab−c))]}^(1/3) p,q = (s/2)±(√((s^2 /4)−m)) t=−(((3pq^2 +2apq+ap^2 +2bp+bq+3c))/((p^3 +ap^2 +bp+c))) x=((pt+q)/(t+1)) . (Please help checking..) (edited a digit 1 in place of 4)

$${Rigorously}\:{over}\:{one}\:{month}'{s} \\ $$$${time},\:{I}\:{developed}\:{a}\:{formula}\:{for} \\ $$$${general}\:{cubic}. \\ $$$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$${let}\:\:{x}=\frac{{pt}+{q}}{{t}+\mathrm{1}} \\ $$$$\boldsymbol{{pq}}=\boldsymbol{{m}},\:\boldsymbol{{p}}+\boldsymbol{{q}}=\boldsymbol{{s}} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\boldsymbol{{m}}^{\mathrm{2}} \left\{\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)^{\mathrm{2}} −\mathrm{6}\boldsymbol{{a}}\left(\boldsymbol{{ab}}−\boldsymbol{{c}}\right)\right\} \\ $$$$+\boldsymbol{{m}}\left\{\mathrm{2}\left(\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}\right)\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)−\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\left(\boldsymbol{{ab}}−\boldsymbol{{c}}\right)\left(\boldsymbol{{ab}}+\mathrm{3}\boldsymbol{{c}}\right)\right\} \\ $$$$\:\:\:+\left(\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}\right)^{\mathrm{2}} −\mathrm{6}\boldsymbol{{bc}}\left(\boldsymbol{{ab}}−\boldsymbol{{c}}\right)=\mathrm{0} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\boldsymbol{{s}}=−\frac{\mathrm{2}}{\mathrm{3}}\left\{\frac{\boldsymbol{{m}}\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)+\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}}{\boldsymbol{{ab}}−\boldsymbol{{c}}}\right\} \\ $$$$\:\:+\left\{\frac{\mathrm{8}}{\mathrm{27}}\left[\frac{\boldsymbol{{m}}\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)+\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}}{\boldsymbol{{ab}}−\boldsymbol{{c}}}\right]^{\mathrm{3}} \right. \\ $$$$\left.\:\:\:−\mathrm{8}\left[\frac{\boldsymbol{{m}}^{\mathrm{3}} +\boldsymbol{{bm}}^{\mathrm{2}} +\boldsymbol{{acm}}+\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{ab}}−\boldsymbol{{c}}}\right]\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\boldsymbol{{p}},\boldsymbol{{q}}\:=\:\frac{\boldsymbol{{s}}}{\mathrm{2}}\pm\sqrt{\frac{\boldsymbol{{s}}^{\mathrm{2}} }{\mathrm{4}}−\boldsymbol{{m}}} \\ $$$$\boldsymbol{{t}}=−\frac{\left(\mathrm{3}\boldsymbol{{pq}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{apq}}+\boldsymbol{{ap}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{bp}}+\boldsymbol{{bq}}+\mathrm{3}\boldsymbol{{c}}\right)}{\left(\boldsymbol{{p}}^{\mathrm{3}} +\boldsymbol{{ap}}^{\mathrm{2}} +\boldsymbol{{bp}}+\boldsymbol{{c}}\right)} \\ $$$$\boldsymbol{{x}}=\frac{\boldsymbol{{pt}}+\boldsymbol{{q}}}{\boldsymbol{{t}}+\mathrm{1}}\:. \\ $$$$\left({Please}\:{help}\:{checking}..\right) \\ $$$$\left({edited}\:{a}\:{digit}\:\mathrm{1}\:{in}\:{place}\:{of}\:\mathrm{4}\right) \\ $$

Question Number 79010 Answers: 1 Comments: 0

lim_(x→0) 3x ln(x) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{3x}\:\mathrm{ln}\left(\mathrm{x}\right)\:? \\ $$

Question Number 79004 Answers: 2 Comments: 3

Question Number 79001 Answers: 0 Comments: 3

In a binomial distribution with mean 4 and the probability of success is 1/3, then the number of trials is

$${In}\:{a}\:{binomial}\:{distribution}\:{with}\:{mean}\:\mathrm{4}\:{and}\:{the}\:{probability}\:{of}\:{success}\:{is}\:\mathrm{1}/\mathrm{3},\:{then}\:{the}\:{number}\:{of}\:{trials}\:{is} \\ $$

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