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

niyazikeklk

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

Subhotosh Khan

Super Moderator
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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

Elite Member
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 $$\displaystyle \lim_{x\to 0}\frac{sin(x)}{x}= 1$$.)

Jomo

Elite Member
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) $$\displaystyle \lim_{x\to a} f(x)= f(a)$$

Jomo

Elite Member
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

Elite Member
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
"$$\displaystyle \lim_{x\to a^-} f(x)= \lim_{x\to a^+} f(x)= f(a)$$.