Mathematics

Problem

Let a)

f(x)=\dfrac{x^{2}-4}{x-2}

for

x<1

f(x)=\sqrt{x-1}

for

x\ge 1

Consider the function

f

defined on the real line by these two rules. (a) Determine the domain of

f

. Express your answer in interval notation. (b) Identify every real number

c

at which

f

is not continuous. For each such

c

, compute

\displaystyle \lim_{x\to c^-}f(x)

and

\displaystyle \lim_{x\to c^+}f(x)

, when they exist. (c) Classify each discontinuity found in part (b) as removable, jump, or infinite (essential). Justify each classification with appropriate limit statements. (d) State all intervals on which

f

is continuous.

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