continuous

(adjective)

Without break, cessation, or interruption; without intervening time.

Related Terms

  • interval

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.
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