Check the points where the constant function $f(x)=k$ is continuous.

Step 1: Understand Continuity Definition

A function $f(x)$ is continuous at a point $x=a$ if the limit of the function as $x$ approaches $a$ exists and is equal to the function's value at $a$. This means three conditions must be met: $f(a)$ must be defined, $\lim_{x \to a} f(x)$ must exist, and these two values must be equal.

Step 2: Evaluate the Function Value

For the given constant function $f(x)=k$, the value of the function at any point $x=a$ is simply $k$. This means the function is defined for all real numbers.

Step 3: Evaluate the Limit

The limit of a constant function as $x$ approaches any value $a$ is always the constant itself. This is because the function's value does not change with $x$.

Step 4: Compare Function Value and Limit

Comparing the function value $f(a)$ and the limit $\lim_{x \to a} f(x)$, we find that both are equal to $k$. This satisfies the condition for continuity.

Step 5: Conclusion

Since the continuity condition holds true for any arbitrary real number $a$, the constant function $f(x)=k$ is continuous at all points in its domain, which is all real numbers.

More from this chapter

• Evaluate the limit as x approaches 0 of (e^(x²) - cos(x)) / (x²).
• Evaluate the limit as n approaches infinity of sum from k=1 to n of n / (n² + k²).
• Evaluate: lim(x->0) (sin(5x) - sin(3x)) / (sin(x)).
• Find the derivative of the function given by f(x)=sin (x²).
• Compute the derivative of 6 x^(100)-x^(55)+x.
• lim_x → 0 csc x (√(2 cos² x + 3 cos x) - √(cos² x + sin x + 4)) is:
• Find the derivative of f(x) = x³ - 3x² + 2x - 5 with respect to x.
• Evaluate: (i) lim _x → 1 (x^(15)-1)/(x^(10)-1) (ii) lim _x → 0 (√(1+x)-1)/(x)

All Limits and Derivatives solutions →