Discuss the continuity of the function f given by f(x)={x,x2, if x≥0 if x<0 $
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Step-by-Step Solution
Step 1: Analyze continuity for x>0
For all values of x>0, the function is defined as f(x)=x. This is a polynomial function, and all polynomial functions are continuous everywhere in their domain. Therefore, f(x) is continuous for all x>0.
Step 2: Analyze continuity for x<0
For all values of x<0, the function is defined as f(x)=x2. This is also a polynomial function, which means it is continuous everywhere in its domain. Therefore, f(x) is continuous for all x<0.
Step 3: Analyze continuity at x=0
To check continuity at x=0, we need to evaluate the left-hand limit, the right-hand limit, and the function value at x=0. The left-hand limit is limx→0−x2=0. The right-hand limit is limx→0+x=0. The function value at x=0 is f(0)=0. Since all three values are equal, the function is continuous at x=0.