Continuous
Definition
A function is continuous if there are no breaks, jumps, or holes in its graph. In other words, you can draw the graph of a continuous function without lifting your pencil.
Analogy
Imagine driving on a smooth road with no potholes or speed bumps. You can smoothly travel from one point to another without any interruptions.
Related terms
Discontinuous: A function is discontinuous if it has breaks, jumps, or holes in its graph.
Limit: The value that a function approaches as the input gets closer and closer to a certain point.
Intermediate Value Theorem: If a continuous function takes on two different values at two points in an interval, then it must also take on every value between those two points within the interval.
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Practice Questions (13)
If a function has a removable discontinuity at x = 2, what can be done to make the function continuous at that point?
Is f(x) = sqrt(x + 4) continuous over the closed interval [-4, 6]?
Is m(x) = 1/(x-2) continuous over the open interval (-1, 3)?
Is g(x) = (x^2 - 1)/(x - 1) continuous over the open interval (-∞, 1)?
Is s(x) = 2x^2 - 3x + 1 continuous over the closed interval [0, 2]?
Is q(x) = e^(3x) continuous over the closed interval [1, 4]?
Is h(x) = cos(2x) continuous over the closed interval [0, π]?
Is k(x) = ln(x + 3) continuous over the open interval (-3, ∞)?
For a piecewise function, what must be true for the function to be continuous at the points where the pieces connect?
If a function is continuous at a point, it must also be:
Which of the following functions is continuous but not differentiable at x = 1?
Which of the following functions is both continuous and differentiable at x = 0?
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), which of the following statements is true?
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