1.3: Continuity

Intermediate Value Theorem: this says that there is always a value between? This doesn't work if the function is discontinuous at a? To find discontinuities, look for values that make the denominator =? Discontinuities are removable as long as the one-sided limits are the? On the AP test, you need to state conditions to use the? Ex. Stage that function x is continous at?

2 values on a graph, point, 0, same, theorem, [a, b]

Continuous function def.? There are no breaks in the? Ex. Of a Break in the graph? To show a continous point at, say, x = 0: say...? This means the point 0 corresponds with a? (On the? Sometimes, you need to show with interval notation that the domain has? Ex. 0<c<4, and c cannot equal 1/3, the lim x approaches c f(x) = f(c)

A function with outputs that vary continously with inputs, graph, a hole, at x = 0 the lim x approaches 0^+ f(x) = f(c), point on the graph, continous points,

Composite continous functions: check if f is continous at? Then plug c into? See if g(x) is continuous at? Given f(x) = sin (x^2 +1). g(x) is? h (x) is? Prove that both of them are? Y = int (x+1), which parts are discontinuous?

C, G, f(c), sin x, x^2 +1, continous functions,all parts x<-1,

A function is continous on an interval it and only if it is? Continous function def.?-doesn't have the be continous at the entire? Rational functikns like y = 1/x is continous at? Point of discontinuity at? Polynomial functions are? Why? Absolute value functions like y =|x| is?

Continous at every point of the interval, continous at every point of its domain, interval, every point of its interval, x =0, continous at every real number c, lim x approaches c f(x) = f(c), continous at every real number,

Logarithmic, exponential, trig, and radical functions are? Theorem 6: properties of composite functions. Sums:? Differences? Products? Constants? Quotients?

Continous at every point of their domain, f+g, f-g, f•g, k•f, f/g,

Given square root x-1/x-1: to prove it is a continuous function, show that it is? The hole x=1 is not part of its? Given f(x) = sin (x^2 +1), to check if it is continuous with theorem 7, find which composite? Say that f(x) is continous because it is a? (Of?)

Continuous at every point of its domain, domain, (g•h) (x), composite of continous functions

Composites: all composites of continous functions are? Ex. (G•f)(x). Theorem 7: If f is continuous at c and g is continous at f(c), then (g•f) (c) is continous at? Given y = |xsin x/x^2 +2|. Identify the? (2) check to see if they are both?

Continuous, c, g(x) /f(x) functions, continous,

For x = c, limit x approaches c must? It must be equal to the value of f at? Open circle on a graph means the point c does not at? Different types of discontinuity: jump discontinuity def.? Ex. Function? Infinite discontinuity? Ex. Function? Oscillating; it moves between? There are no?

Exist, x = c, f (c), when one-sides limits don't approach the same value, piecewise, when the function increases/decreases without Bound, rational, the same values, limits,

To make the function continuous at point x=a, set it equal to the? Ex. Open circle at (0,0). Make f(-1) =? For describing the intervals and points of discontinuity, say that the function is discontinuous at? (2)

Function value, 0, all points outside the domain/ and all values x=a,

Vertical asymptotes of tan, cot, csc, and sec are which type of discontinuity? Limits can exist at? (If and only if?) if there is an open and closed circle point at the same x value: are they included in the domain? For interval notation, use? Given (0,1) is an open circle, to make the function continous at x =0, set f(0) =?

Infinite, open circle points if and only if the left and right limits are the same, no,( ), 1,

For discontinuous points like x =1, write it like...? Some discontinuous points have a? (On the?) but it does not? Ex. At x = 2, lim x approaches 2 f(x) =1, but to be continous, it would have to equal? For areas outside the geaph's domain, create interval notation to show they are?

Lim x approaches 1 f(x) does not exist, point on the graph, exist, f(2), discontinuous

Interior point equation: the function is continous at a point c if...? Continous Endpoint equation: left equation? Right? Discontinuous point: write that c is a point of? (At?) this in the domain of?

Lim x approaches c f(x) = f(c), lim x approaches a^+ f(x) = f(a), lim x approaches b^- f(x) = f(b), discontinuity at f, f,

Given a point of discontinuity x = 2 and the left limit is 3 and the right limit is 2, can it be removed? If there is no limit at this point, it cannot be? To find the g(x) extended function: first, use? The final equation like (x+2) is? How do you find the lim x approaches 2 (x+2)?

No, removed, synthetic division, g(x), direct substitution,

For function and finding continous interval: find the? Ex. [-1, 3). For your answer, write What? Since 3 is excluded, it is not a? To make x = a a removable discontinuity, you have to make it equal to the? For trig and int(x) functions in problems about the extending function, write it like a? Ex. G(x) = {sin 4x/x, x cannot equal 0 and 4, x =0}. The piecewise includes the? (2)

Overall domain, f is continuous at all points of the domain [-1, 3) except x =1, point of discontinuity, function value, piecewise function, original function/the value at x=a,

Given (4-x^2, x<1) and (ax^2-9, 1), what do you do? Ex. a-9 = 3. Solve for? A function is continous at c if and only if: three conditions for continous points? (3) there can't be any jump? To find the limit, first find the? (2) For removable discontinuity: if there is an open circle and shaded circle for the same x value, it is not? The left and rights limits will be? Equation to show a function is continuous?

Plug in 1 to find a in ax^2-9/ plug 1 into 4-x^2 to find the limit value/ find a by making the a equation = limit value, a, f(c) function value exists/lim x approaches c exists/ lim x approaches c = f(c) function value, discontinuities, left/right limits, removable, different, lim x approaches c^- f(x) = lim x approaches c^+ f(x) = f(c),

Given (x-3)^2/ (x-3)^3, if the hole remains in the denominator, it is not? For (x-2)^2/ (x-2), why is it removable? To set x =0 to f(0) to make it continuous, what is the condition for the point? (2)

Removable, the hole is removed from the denominator, the left/right-sided limit must be the same,

To remove a discontinuity, use?Synthetic division: it uses which equation? Ex. (X +5): the value is? Pull down the? For functions like x^3-7x-6, substitute the x^2 term with? Pull down the first? Multiply by the? Put it under the second? Add the? Then? Then, write a new function called? In this: write it like a? Ex. X =3 and g (x) = {x^3-7x-6, x cannot equal 3 {10/3, x =3). For this 2nd part, plug in the x =a value after removing the? For remainder values, write it as a ratio of?

Synthetic division, (x-a), -5, coefficients, 0, coefficient, a, coefficient, values, repeat, g(x), piecewise function, hole, remainder/x-a

For lim x approaches 0 (-sinx/4x) and g(x) = -sin3x/5x, x cannot equal 0 and ____, x =0. To find the limit value at 0, create a? Plug in? # of decimals in AP CALC?

Table, values, 3 decimals,

If f(x) is continuous at c, then lim x approaches f(x) = f(c) is? F(c)? Lim x approaches 0 f(x)?

True, exists, exists,

Word problem: show that Y= 36,500 (1.035)^0, (1.035)^1, (1.035)^2 = this shows that? Given square root x-1, the value x=1 is not a? This is because it is excluded from the?

Y= 36,500 (1.035)^int (x), discontinuity, domain