Let f be a real-valued function whose domain is a subset of R. Then f is continuous at x_0 in dom(f) if and only if for each e > 0 there exists δ > 0 such that x ∈ dom(f) and |x − x_0| < δ imply |f(x) − f(x0)| < e. Use the defintion above to prove that: f(x) = x(x-1) if x is rational                                                                       = 0 if x is irrational  is continuous at x = 0 and x =1 and discontinuous everywhere els

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: = Complete the following truth table. Use T for true and F for false. You may add more columns, but…

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Q: f(x) = x² - 2, x + 4, O true false x 3 is continuous everywhere.

A: f(x) = ( x2-2, x&lt;=3            x+4, x&gt;3

Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x&lt;1           x…

Q: Consider the function f with domain (-2, : 3|22 – 1| if – 2 < x < 1 2x2 + 2x – 4 f(x) = 3 if x = 1…

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Q: Let f: X --> Y be a function, and let U be a subset of X.  (1) Prove or disprove that f maps X \ U…

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Q: Give an example of a function that is continuous on (-∞,0) U (0,1] U [1,0). se limits and theory…

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Q: Show that if f"(x) < 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] < favg < f[(a+b)/2]

A: Consider the given information. f"(x) &lt; 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] &lt; favg &lt;…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.