Question

Discuss the continuity of each function. $$ f(x)=\left\\{\begin{array}{ll} x, & x<1 \\ 2, & x=1 \\ 2 x-1, & x>1 \end{array}\right. $$

Short answer

The function is continuous for all \(x \neq 1\), but discontinuous at \(x = 1\).

Step-by-step solution

Calculate the limits from left and right

Compute the limit of the function as \(x\) approaches 1 from the left (\(x<1\)) and from the right (\(x>1\)). The left-hand limit is \( \lim_{{x \to 1}^-} f(x) = 1\) and the right-hand limit is \( \lim_{{x \to 1}^+} f(x) = 1\).

Compare the limits with the function value at the point

The function value at \(x = 1\) is \(f(1) = 2\). The value of the function at the point \(x=1\) does not equal the left-hand or right-hand limits.

Draw the conclusion

A function is continuous at a point if the left-hand limit, the right-hand limit, and the function value at the point are all equal. In this case, as \(f(1)\) is not equal to the limits as \(x\) approaches 1 from left and right, \(f(x)\) is not continuous at \(x=1\). However, for all other values of \(x\), the function is continuous.