# Over what intervals are the following functions continuous?

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Over what intervals are the following functions continuous? Justify your answer using the definition of continuity.

a.

b.

Let f and g be twice differentiable functions such that for all x in the domain of f. If and What (if anything) can you say about f, g, and h at x = 2?

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

a), h(x) is continuous for all real x values based on the fact that superposition of elementary functions is continuous function everywhere where the components exist and are continuous functions (we must consider domains for all the components). x^2 exists for all real x, sin(x) exists for all real x, and these elementary functions are continuous. Thus sin(x^2) is continuous function for all real x.

Similar situation for k(x). Square root of (x^2 + 7) exists for all x because (x^2 + 7) > 0. Division does not exists when denominator is equal to zero. This corresponds to x = +/- 3 ( for these x values x^2 +7 = 16 and denominator is zero).

Thus k(x) is continuous everywhere except x = +/- 3.

h'(x) = f ' (g(x)) * g ' (x) . h ' (2) = 4, so h ' (2) > 0. Because f ' (x) <= 0 we must have g ' (2) < 0. Thus, the result is the following.

g(x) is decreasing function at x = 2, f(x) is decreasing function at x = 2, but h(x) is increasing function at x = 2.

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