Consider the function ( f ) given by [ f(x)=left{egin{array}{l} g(x) ext { if } x leq 0 \ h(x) ext { if } x>0 end{array} ight. ] These functions are depicted below. Note that ( f ) is discontinuous at ( x=0 ). By clicking and dragging the empty dot of ( h(x) ) make ( f ) into a continuous function on ( mathbb{R} )
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Step 1
First, we need to understand what it means for a function to be continuous. A function is continuous at a point if the limit of the function as x approaches that point from both sides is equal to the value of the function at that point. Show more…
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Key Concepts
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Consider the functions $$ f(x)=\left\{\begin{array}{ll}{1,} & {0 \leq x} \\ {0,} & {x<0}\end{array} \text { and } g(x)=\left\{\begin{array}{ll}{0,} & {0 \leq x} \\ {1,} & {x<0}\end{array}\right.\right. $$ In each part, is the given function continuous at $x=0 ?$ $$ \begin{array}{llll}{\text { (a) } f(x)} & {\text { (b) } g(x)} & {\text { (c) } f(-x)} & {\text { (d) }|g(x)|} \\ {\text { (e) } f(x) g(x)} & {\text { (f) } g(f(x))} & {\text { (g) } f(x)+g(x)}\end{array} $$
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