Let f(x) = \left\{\begin{matrix} x^2 - 5 if x less than or equal to a \\ x + 1 if a less than x...

Question:

Let {eq}f(x) = \left\{\begin{matrix} x^2 - 5 \,\,\,\,\,\,\, if \,\, x \leq a \\ x + 1 \,\,\,\,\,\,\,\,\, if \,\, a < x \end{matrix}\right. {/eq}

Find the value(s) of {eq}a {/eq} for which {eq}f(x) {/eq} is continuous everywhere.

Continuous Functions :

Let f be a real function on a subset of the real numbers and let c be a point in the

domain of f. Then f is continuous at c if {eq}\lim _{x \to c }f(x)=f(c) {/eq}

More elaborately, if the left hand limit, right hand limit and the value of the function

at x = c exist and are equal to each other,

{eq}\lim _{x \to c^{-} }f(x)=f(c)=\lim _{x \to c^{+} }f(x){/eq}

then f is said to be continuous at {eq}x=c {/eq}

Answer and Explanation: 1

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Let {eq}f(x) = \left\{\begin{matrix} x^2 - 5 \,\,\,\,\,\,\, if \,\, x \leq a \\ x + 1 \,\,\,\,\,\,\,\,\, if \,\, a <...

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Continuous Functions Theorems

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Chapter 5 / Lesson 10
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Continuous functions are functions that have their conditions satisfied between multiple points, appearing as an uninterrupted line when graphed. See examples of how this is represented in the intermediate value theorem and the extreme value theorem.


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