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I don’t know how to do this

I don’t know how to do this-example-1
User Elissa
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1 Answer

5 votes

To check for continuity at the edges of each piece, you need to consider the limit as
x approaches the edges. For example,


g(x)=\begin{cases}2x+5&\text{for }x\le-3\\x^2-10&\text{for }x>-3\end{cases}

has two pieces,
2x+5 and
x^2-10, both of which are continuous by themselves on the provided intervals. In order for
g to be continuous everywhere, we need to have


\displaystyle\lim_(x\to-3^-)g(x)=\lim_(x\to-3^+)g(x)=g(-3)

By definition of
g, we have
g(-3)=2(-3)+5=-1, and the limits are


\displaystyle\lim_(x\to-3^-)g(x)=\lim_(x\to-3)(2x+5)=-1


\displaystyle\lim_(x\to-3^+)g(x)=\lim_(x\to-3)(x^2-10)=-1

The limits match, so
g is continuous.

For the others: Each of the individual pieces of
f,h are continuous functions on their domains, so you just need to check the value of each piece at the edge of each subinterval.