1. Which of the above functions are not continuous where they are defined? Why?
2. Which of the above functions are not differentiable where they are defined? Why?
Photo Link: goo.gl/photos/2RPU1dAWoKzUn4Pf8
For the first question, I said that graph F is not continuous because of a jump, but I'm missing another one from those 6 graphs.
For the second question, I said b is not differentiable because of corners, but I appear to be missing 3 more. I initially answered C, D, and F but those aren't correct.
(All 6 graphs are part of the questions and A and a are two different graphs fyi)
Any help is greatly appreciated!
For continuous, ya got me. Only F looks like it fails.
For differentiable, surely F fails, because it isn't even continuous. Also, (a),(b),(d) fail because of the cusps. I am assuming that d(0) is defined somewhere on the y-axis.
(f) fails because the ends of the semi-circle have vertical tangents.
To determine which functions are not continuous and which functions are not differentiable from the given graphs, we need to examine the definition and conditions for continuity and differentiability.
1. Continuity:
A function is said to be continuous at a point if the limit of the function exists and is equal to the value of the function at that point. In other words, there should be no abrupt jumps, holes, or breaks in the graph.
Looking at the provided photo link, you mentioned that graph F has a jump, but you are missing another one. To identify the points of discontinuity, examine each graph closely and look for any points where the graph has a vertical or horizontal gap, or where the graph has a removable or non-removable jump or hole.
To give you a general idea:
- Graph A: It appears to be a continuous curve without any abrupt jumps or breaks. Hence, it is continuous where it is defined.
- Graph a: Without viewing the graph, it is challenging to identify its continuity status.
For the graphs you haven't mentioned, carefully analyze each one:
- Graph B: Check if there are any jumps, breaks, or other discontinuities in the graph.
- Graph C: Similar to graph B, analyze the graph for any discontinuities.
- Graph D: Also, inspect for any discontinuous behavior.
- Graph E: Determine if there are any abrupt changes or gaps in the graph.
- Graph F: You have already mentioned that there is a jump, but ensure you haven't missed any other discontinuities.
2. Differentiability:
A function is said to be differentiable at a point if its derivative exists at that point. In other words, the function should have a well-defined tangent line without any sharp corners, vertical tangents, or jumps in the derivative.
You mentioned that graph b is not differentiable due to corners, but you are missing three more. Look for graphs where the derivative shows any sharp corners, vertical tangents, or jumps.
Similarly, examine the remaining graphs:
- Graph A: To determine its differentiability, find the derivative and inspect it for any abnormalities.
- Graph a: Without viewing the graph or having additional information, it is challenging to establish its differentiability.
For the graphs you haven't mentioned, analyze each one for any peculiarities in the derivative:
- Graph C: Compute the derivative and examine for any irregularities.
- Graph D: Similarly, find the derivative and assess its behavior.
- Graph E: Calculate the derivative and analyze it for any deviations.
- Graph F: You already identified its non-differentiability due to corners, but ensure you have not missed any other specific peculiarities.
Remember that without inspecting the actual graphs, it is difficult to provide precise answers. Therefore, I encourage you to closely examine each graph and apply the concepts of continuity and differentiability to make accurate determinations.