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?

 

[Subhotosh Khan]

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

 

[Jomo]

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]

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