Prove the continuity of $\hat{f}$ without the Tube Lemma

I am currently in a point-set topology course, and on a past homework assignment we were asked to prove the following:

If $ X$ is a locally compact space, then $ f$ is continuous if and only if $ \hat{f}$ is continuous.

Where the definition of locally compact that we are using is “A space $ X$ is locally compact at $ x\in X$ , if every neighborhood of $ x$ contains a compact subset $ K$ with $ x\in int(K)$ .”

Additionally, $ f: X\times Y\to Z$ is continuous, we define $ \hat{f}: Y\to\mathcal{C}(X,Z)$ by $ \hat{f}(y)(x)=f(x,y)$ .

The backwards direction was easy to prove, but I had a lot of trouble proving the forward direction. After the due date, our professor sent out a proof making use of the Tube Lemma (which we had not yet covered or mentioned). With the Tube Lemma it now seems pretty straightforward, but I’m now curious as to how to prove the forward direction without the Tube Lemma, which I believe to be possible (but probably harder).

Would someone be willing to outline a proof of the forward direction without the Tube Lemma, or at least mention the main idea of the proof?

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