Question

True or False For any function $f$, and any real number c,$\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)$.

Short answer

The statement is false.

Step-by-step solution

Step. 1 Given Information

It is given in the question that $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)$, at any c.

i.e., Left hand limit is equal to right hand limit for the function $f\left(x\right)atx=c.$

Step. 2 Example

Let suppose this function,

$f\left(x\right)=\left\{\begin{array}{l}2x+1,x<0\\ 0,x⩾0\end{array}\right.$

Now, $$\begin{gathered} \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{-}}{\lim }(2x+1)=2\left(0\right)+1=1, \\ \underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }0=0, \end{gathered}$$

We can see clearly from this example that LHL is not equal to RHL.

Step. 3 Conclusion

From above example it is clear that there exist atleast one function for which LHL is not equal to RHL. Therefore, the above statement is false.