The graph of 𝑓 is shown in the figure. At which point is 𝑓 continuous but not differentiable?
Let’s look in turn at each of the points 𝑎, 𝑏, 𝑐, and 𝑑 on our graph of our function and establish what we do know about them. It seems sensible to begin with the point 𝑎. We can see we have this vertical dash line. This line will have the equation 𝑥 equals 𝑎, where 𝑎 is some constant less than zero. If we look carefully at the graph of our function, we see that the line tends to 𝑥 equals 𝑎 from the left and never quite reaches it. It also appears to do the same from the right. And this implies that the line 𝑥 equals 𝑎 is an asymptote.
More importantly, we have a discontinuity here. It appears to be an asymptotic discontinuity though the type of discontinuity doesn’t actually matter. We’re looking for a point on our curve which is continuous. So it’s absolutely can’t be continuous at this point. So we’ll move on to point 𝑏. It’s clearly continuous here. There are no jumps or holes in the curve. There’s also no reason to believe it’s not differentiable at this point either. We could add a tangent to the curve at this point. And this will allow us to easily find the rate of change of our function with respect to 𝑥 at this point. So it can’t be 𝑏. 𝑓 is continuous and differentiable at this point.
We’ll now look at 𝑐. It does appear to be continuous though here we have this sharp turn in our graph. This means 𝑓 can’t be differentiable at 𝑐. We can’t draw a unique tangent to the curve at this point. And this means 𝑓 must therefore be continuous, but not differentiable at 𝑐. We will double check point 𝑑. We can see that we have a discontinuity. This open circle tells us that the function is not defined at this point. And so it can’t be continuous here.
𝑓 is continuous and not differentiable at the point 𝑐.