For the given function f(x) below, find the constants
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What does it mean when a function f(x) is continuous at x = xo?
What does it mean when a function f(x) is differentiable at x = xo?
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HallsofIvy
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Do you know what "continuous" and "differentiable" mean?
The function f(x) is continuous at x= a if and only if \(\displaystyle \lim_{x\to a} f(x)= f(a)\).
For a piecewise defined function like this, the two one-sided limits must the same so we must have \(\displaystyle \lim_{x\to a^+} f(x)= \lim_{x\to a^-} f(x)= f(a)\).
The derivative of f(x) at x= a is \(\displaystyle \lim_{h\to 0} \frac{f(a+h)- f(a)}{h}\) so in order that f be differentiable, that limit must exist.
What is f(1)?
The function f(x) is continuous at x= a if and only if \(\displaystyle \lim_{x\to a} f(x)= f(a)\).
For a piecewise defined function like this, the two one-sided limits must the same so we must have \(\displaystyle \lim_{x\to a^+} f(x)= \lim_{x\to a^-} f(x)= f(a)\).
The derivative of f(x) at x= a is \(\displaystyle \lim_{h\to 0} \frac{f(a+h)- f(a)}{h}\) so in order that f be differentiable, that limit must exist.
What is f(1)?