Continuity Problem: Solutions & Resources

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SUMMARY

A function is continuous at a point if its limit at that point equals its function value. In the discussion, it is established that for the function \( f(x) \), both the limit as \( x \) approaches 1 and the function value at \( x=1 \) are equal to 0, confirming continuity at that point. The conversation further explores the continuity of \( f \) at other integer values and the existence of real values where \( f \) may not be continuous, indicating a broader investigation into the function's behavior.

PREREQUISITES
  • Understanding of limits in calculus
  • Knowledge of continuity definitions in mathematical analysis
  • Familiarity with left-hand limits (lhl) and right-hand limits (rhl)
  • Basic function evaluation techniques
NEXT STEPS
  • Research the properties of continuous functions in calculus
  • Learn about discontinuities and types of discontinuities in functions
  • Explore the concept of limits at infinity and their implications
  • Study the Intermediate Value Theorem and its relation to continuity
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Students of calculus, mathematics educators, and anyone studying the properties of functions and their continuity.

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A function is continuous at a point if its limit at that point is the same as its function value.

What is $\lim_{x\to 1} f(x)$ and what is $f(1)$?
 
I'm getting $f(1)$ as well as $\lim_{x \to 1}f(x)$ = 0 both lhl and rhl
 
DaalChawal said:
I'm getting $f(1)$ as well as $\lim_{x \to 1}f(x)$ = 0 both lhl and rhl
Then $f$ is continuous at $x=1$ and we have eliminated answer a.

The remaining question is what happens at other values of $x$.
Can we find another integer value for $x$ where $f$ is continuous?
Can we find a real $x$ where $f$ is not continuous?
 

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