What must a and b be for the function to be continuous on R?

[niyazikeklk]

What must a and b be for the function to be continuous on R?

 

[Deleted member 4993]

View attachment 21637

Fonksiyonun R'de sürekli olabilmesi için a ve b ne olmalıdır?

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Hint: How would you prove that a function is continuous at a given point?

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[HallsofIvy]

A function, f(x), is continuous at x= a if and only if
a) $\displaystyle \lim_{x\to a^+} f(x)$ exists
b) $\displaystyle \lim_{x\to a^-} f(x)$ exists
c) $\displaystyle \lim_{x\to a^+} f(x)= \lim_{x\to a^-} f(x)$.

So what are $\displaystyle \lim_{x\to a^+} f(x)= \lim_{x\to 0} 0$ and $\displaystyle \lim_{x\to a^-} f(x)= \lim_{x\to 0} (x^2)^a sin^b(x^2)$?

(It helps to know that $\displaystyle \lim_{x\to 0}\frac{sin(x)}{x}= 1$.)

 

[Steven G]

A function, f(x), is continuous at x= a if and only if
a) $\displaystyle \lim_{x\to a^+} f(x)$ exists
b) $\displaystyle \lim_{x\to a^-} f(x)$ exists
c) $\displaystyle \lim_{x\to a^+} f(x)= \lim_{x\to a^-} f(x)$.

So what are $\displaystyle \lim_{x\to a^+} f(x)= \lim_{x\to 0} 0$ and $\displaystyle \lim_{x\to a^-} f(x)= \lim_{x\to 0} (x^2)^a sin^b(x^2)$?

(It helps to know that $\displaystyle \lim_{x\to 0}\frac{sin(x)}{x}= 1$.)

Prof, what you stated above via a-c only shows that the limit at x=a exists. To show continuity at x= a I am sure that you know that you also need to have d) $\displaystyle \lim_{x\to a} f(x)= f(a)$

 

[Steven G]

I do not think that this problem has an answer as stated. You can however answer what can a+b be for continuity at x=0

 

[HallsofIvy]

Prof, what you stated above via a-c only shows that the limit at x=a exists. To show continuity at x= a I am sure that you know that you also need to have d) $\displaystyle \lim_{x\to a} f(x)= f(a)$

Yes, the last line should have been
"$\displaystyle \lim_{x\to a^-} f(x)= \lim_{x\to a^+} f(x)= f(a)$.