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BCC.01.9 – Continuity and Differentiability of Functions

BCC.01.9 – Continuity and Differentiability of Functions. MCB4U - Santowski. (A) Continuity. We can introduce another characteristic of functions  that of continuity. We can understand continuity in several ways:

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BCC.01.9 – Continuity and Differentiability of Functions

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  1. BCC.01.9 – Continuity and Differentiability of Functions MCB4U - Santowski

  2. (A) Continuity • We can introduce another characteristic of functions  that of continuity. We can understand continuity in several ways: • (1) a continuous process is one that takes place gradually, smoothly, without interruptions or abrupt changes • (2) a function is continuous if you can take your pencil and can trace over the graph with one uninterrupted motion

  3. (B) Conditions for Continuity • a fcn is continuous at a given number, x = a, if: • (i) f(a) exists; • (ii) exists • (iii) • In other words, if I can evaluate a function at a given value of x = a and if I can determine the value of the limit of the function at x = a and if we notice that the function value is the same as the limit value, then the function is continuous at that point. • So a function is continuous over its domain if it is continuous at each point in its domain.

  4. (I) Jump Discontinuities: ex and it’s limit and function values at x = 1. We notice our function values and our limits (LHL and RHL) "jump" from 4 to 0 (C) Types of Discontinuities

  5. (II) Infinite Discontinuities ex. and it’s limit and function values at x = 0. The left hand limit and right hand limits are both infinite although the function value is 1 (C) Types of Discontinuities

  6. (III) Removable Discontinuities Ex and it’s limit and function values at x = 2. The left hand limit and right hand limits are equal to 3 although the function value is 1 (C) Types of Discontinuities

  7. (D) Examples • Find all numbers, x = a, for which each function is discontinuous. For each discontinuity, state which of the three conditions are not satisfied. • (i) (ii) • (iii) • (iv)

  8. (E) Continuity and Differentiability

  9. (E) Continuity and Differentiability • Recall the fundamental idea that a derivative at a point is really the idea of a limiting sequence of secant slopes (or tangent line) drawn to a curve at a given point • Now , if a function is discontinuous at a given point, try drawing secant lines from the left and secant lines from the right and then try drawing a specific tangent slope at the point of discontinuity in the following diagrams • Conclusion  you can only differentiate a function where is it is continuous

  10. (F) Continuity and Differentiability – Online Examples • Follow this link to One-sided derivatives from IES Software • And then follow this link to Investigating Differentiability of Piecewise Functions from D. Hill (Temple U.) and L. Roberts (Georgia College and State University

  11. (G) Continuity and Differentiability • One other point to add that comes from our study of the last two examples  even if a function is continuous, this does not always guarantee differentiability!!!! • If a continuous function as a cusp or a corner in it, then the function is not differentiable at that point  see graphs on the next slide and decide how you would draw tangent lines (and secant lines for that matter) to the functions at the point of interest (consider drawing tangents/secants from the left side and from the right side) • As well, included on the graphs are the graphs of the derivatives (so you can make sense of the tangent/secant lines you visualized)

  12. (G) Continuity and Differentiability – Examples • Continuous functions are non-differentiable under the following conditions: • The fcn has a “corner” (ex 1) • The fcn has a “cusp” (ex 2) • The fcn has a vertical tangent (ex 3) • This non-differentiability can be seen in that the graph of the derivative has a discontinuity in it!

  13. (H) Internet Links for Continuity • Calculus I (Math 2413) - Limits - Continuity from Paul Dawkins • A great discussion plus graphs from Stefan Waner at Hofstra U  Continuity and Differentiability  then do the Continuity and Differentiability Exercises on this site • Here are a couple of links to Visual Calculus from UTK • General discussion plus examples and explanations: Continuous Functions • Quiz to take on continuous functions: Continuity quiz • And a second, different type of quiz: Visual Calculus - Drill - Continuity of Piecewise Defined Functions

  14. (I) Homework • Handouts from Stewart, 1997

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