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Help with piecewise functions?
Here is the problem I am having difficulty with:
s(x) = { k^2 + x if x=<(equal or less than) k, x^2 + k if x > k
Show that s(x) is a continuous function for every value of k E R (K is and element of the real numbers)
Thanks in advance!
1 Answer
- KarnageLv 58 years agoFavorite Answer
To show if a function is continuous, you must show that:
limit as x → a of f(x) = f(a)
Obviously, s(x) is continuous for x < k and x > k since k^2 + x and x^2 + k are everywhere-continuous polynomials. All that's required to check is when x = k.
The limit:
limit as x → k of f(x)
exists when both left and right side limits agree.
Right side: limit as x → k+ of f(x) = k^2 + k [here, we substitute x = k into x^2 + k]
Left side: limit as x → k- of f(x) = k^2 + k [here, we substitute x = k into k^2 + x]
Since left and right side limits are the same, the function is continuous.
Hope this helps!
