continuous

Examples of continuous in the following topics:

• The Intermediate Value Theorem

  • For each value between the bounds of a continuous function, there is at least one point where the function maps to that value.
  • are continuous - there are no singularities or discontinuities.
  • The function is defined for all real numbers x ≠ −2 and is continuous at every such point.
  • The question of continuity at x = −2 does not arise, since x = −2 is not in the domain of f.
  • There is no continuous function F: R → R that agrees with f(x) for all x ≠ −2.

• Piecewise Functions

  • The domain of the function starts at negative infinity and continues through each piece, without any gaps, to positive infinity.  
  • Since there is an closed AND open dot at $x=1$ the function is piecewise continuous there.  
  • When $x=2$, the function is also piecewise continuous.  
  • The range begins at the lowest $y$-value, $y=0$ and is continuous through positive infinity.  

• Interest Compounded Continuously

  • This might not seem like a lot but the amount of interest earned will continue to increase each year as there is more and more money in the account.
  • You earn the most interest when interest is compounded continuously.
  • We expect that as $n$ increases the amount in the account also increases, but if the amount grows without bounds then banks would be giving away much money as they compound interest continuously.
  • The formula for compound interest with the number of compounding periods going to infinity yields the formula for compounding continuously.
  • The graph shows that the more frequent the number of compounding periods the more interest is accrued and shows this visually for yearly, quarterly, monthly and continuous compounding.

• Visualizing Domain and Range

  • Both graphs include all real numbers $x$ as input values, since both graphs continue to the left (negative values) and to the right (positive values) for $x$ (inputs).  
  • The curves continue to infinity in both directions; therefore, we say the domain for  both graphs is the set of all real numbers, notated as: $\mathbb{R}$.

• The Substitution Method

  • Continue until you have reduced the system to a single linear equation.

• Standard Equations of Hyperbolas

  • Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does.

• Zeroes of Polynomial Functions With Rational Coefficients

  • The decimal expansion of an irrational number continues forever without repeating.

• Basic Operations

  • Continuing the previous example, say you start with a group of 5 boxes.

• The Number e

  • .$ By then asking about what happens as $n$ gets arbitrarily large, he was able to come up with the formula for continuously compounded interest, which is $A=Pe^{rt}

• Population Growth

  • A high estimation predicts the graph to continue at an increasing rate, a medium estimation predicts the population to level off, and a low estimation predicts the population to decline.